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ELEMENTARY 
MATHEMATICAL  ANALYSIS 


MODERN  MATHEMATICAL  TEXTS 

EDITED  BY 

Charles  S.  Slighter 


ELEMENTARY  MATHEMATICAL  ANALYSIS 
By  CHABLsa  S.  Slichteb 
i97  pages,  5%  7ii,niiiatraled $2,60 

MATHEMATICS  FOR  AGRICULTURAL 
STUDENTS 
By  Henry  C.  Wolff 
311  panes,  5  %  7H,  lUustraled .  $1.50 

CALCULUS 
By  Herman  W.  March  and  Henry  C.  Wolff 
360  pages,  5  I  7H,  Illustrated.    .  $2.00 

PROJECTIVE  GEOMETRY 
,  By  L.  Watland  Dowling 

316  pages,  a  ii7H,IllustraUd $2.00 


MODERN    MATHEMATICAL    TEXTS 
Edited  bt  Chaelbs  S.  Slichtee 

ELEMENTARY 
MATHEMATICAL  ANALYSIS 

A   TEXT  BOOK  FOR  FIRST 
YEAR  COLLEGE  STUDENTS 


BY 
CHARLES  S.  SLICHTER,  Sc.  D. 

PROFESSOR  OF  APPLIED    MATHEMATICS 
■DNIVERSITY    OP  WISCONSIN 


Second  Edition 
Revised  and  Entirely  Reset 


McGRAW-HILL  BOOK  COMPANY,  Inc. 
.  239  WEST  39TH  STREET.    NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  Ltd. 

6  &  8  BOUVBKIE  ST.,  E.  C- 

1918 

A- 


COPYEIGHT,    1914,    1918,   BY  THe] 

McGeaw-Hill  Book  Company,  Inc. 


TMT-    MAPtK    PUESS    roBK    FA, 


PREFACE  TO  THE  SECOND  EDITION 


In  rewriting  the  present  book,  simplification  of  the  material 
has  been  the  main  end  in  view.  Considerable  matter  has  been 
omitted,  and  numerous  worked  exercises  have  been  added.  The 
second  chapter  is  devoted  to  an  introduction  to  rectangular 
coordinates  and  to  the  straight  line.  New  sets  of  exercises  and 
long  lists  of  miscellaneous  and  review  exercises  have  been  inserted 
at  appropriate  places.  Changes  in  order  of  material  and  in 
method  of  treatment  have  been  made  freely. 

Much  greater  use  has  been  made  of  fine  print  than  in  the  first 
edition.  Sections  of  the  text  which  can  readily  be  omitted  have  a 
star  attached  to  the  section  numbers.  Some  of  these  of  a  sub- 
ordinate illustrative  character,  or  primarily  intended  for  reference, 
are  put  in  fine  print. 

The  review  chapter  on  elementary  algebra  has  been  greatly 
enlarged.  This  material  is  placed  in  the  last  chapter  or  appendix, 
where  the  considerable  amount  of  very  elementary  mathematics 
will  not  at  once  confront  and  perhaps  discourage  the  well-prepared 
student.  At  the  same  time  enough  material  is  given  so  that 
students  with  but  a  single  year  of  high  school  algebra  can  be 
gotten  ready  for  the  course.  The  elementary  material  is  so 
classified  that  either  a  few  days,  or  several  weeks  may  be  devoted 
to  the  review. 

The  writer  is  greatly  indebted  to  many  persons  for  aid  in  the 
revision.  Professors  March  and  Wolff  of  the  University  of 
Wisconsin  have  contributed  much,  and  Professor  Wolff  has  read 
all  of  the  galley  proof.  I  am  especially  indebted  to  Professors 
Jordan  and  Lefschetz  of  the  University  of  Kansas  for  many 
valuable  suggestions  and  to  Professor  L.  C.  Plant  of  Lansing, 
Michigan.    To  all  of  these  my  especial  thanks  are  due. 

Charles  S.  Slichter. 


FROM  THE  PREFACE  TO  THE 
FIRST  EDITION 


This  book  is  not  intended  to  be  a  text  on  "Practical  Mathe- 
matics" in  the  sense  of  making  use  of  scientific  material  and  of 
fundamental  notions  not  already  in  the  possession  of  the  student, 
or  in  the  sense  of  making  the  principles  of  mathematics  secondary 
to  its  technique.  On  the  contrary,  it  has  been  the  aim  to  give 
the  ftmdamental  truths  of  elementary  analysis  as  much  promi- 
nence as  seems  possible  in  a  working  course  for  freshmen. 

The  emphasis  of  the  book  is  intended  to  be  upon  the  notion  of 
functionality.  Illustrations  from  science  are  freely  used  to  make 
this  concept  prominent.  The  student  should  learn  early  in  his 
course  that  an  important  purpose  of  mathematics  is  to  express  and 
to  interpret  the  laws  of  actual  phenomena  and  not  primarily  to 
secure  here  and  there  certain  computed  results.  Mathematics 
might  well  be  defined  as  the  science  that  takes  the  broadest  view  of 
all  of  the  sciences — an  epitome  of  quantitative  knowledge.  The 
introduction  of  the  student  to  a  broad  view  of  mathematics  can 
hardly  begin  too  early.  , 

The  ideas  explained  above  are  developed  in  accordance  with  a 
two-fold  plan,  as  follows : 

First,  the  plan  is  to  group  the  material  of  elementary  analysis 
about  the   consideration  of  the  three  fundamental  functions: 

1.  The  Power  Function  y  =  ax"  (n  any  number)  or  the  law 
"as  X  changes  by  a  fixed  multiple,  y  changes  by  a  fixed  multiple  also." 

2.  The  Simple  Periodic  Function  y  =  asin  mx,  considered  as 
fundamental  to  all  periodic  phenomena. 

3.  'The  Exponential  Function,  or  the  law  "as  x  changes  by  a  fixed 
increment,  y  changes  by  a  fixed  multiple." 

Second,  the  plan  is  to  enlarge  the  elementary  functions  by  the 
development  of  the  fundamental  transformations  applicable  to 
these  and  other  functions.     To  avoid  the  appearance  of  abstruse- 


Viii  PREFACE 

ness,  these  transformations  are  stated  with  respect  to  the  graphs 
of  the  functions;  that  is,  they  are  not  called  transformations,  but 
"motions"  of  the  loci.  The  facts  are  summarized  in  several 
simple  "Theorems  on  Loci,"  which  explain  the  translation,  rota- 
tion, shear,  and  elongation  or  contraction  of  the  graph  of  any 
function  in  the  xy  plane. 

Combinations  of  the  fundamental  functions  as  they  actually 
occur  in  the  expression  of  elementary  natural  laws  are  also  dis- 
cussed and  examples  are  given  of  a  type  that  should  help  to  ex- 
plain their  usefulness. 

Emphasis  is  placed  upoji  the  use  of  time  as  variable.  This 
enriches  the  treatment  of  the  elementary  functions  and  brings 
many  of  the  facts  "analytic  geometry"  into  close  relation  to 
their  application  in  science.  A  chapter  on  waves  is  intended  to 
give  the  student  a  broad  view  of  the  use  of  the  trigonometric  func- 
tions and  an  introduction  to  the  application  of  analysis  to  periodic 
phenomena. 

It  is  difficult  to  understand  why  it  is  customary  to  introduce 
the  trigonometric  functions  to  students  seventeen  or  eighteen  years 
of  age  by  means  of  the  restricted  definitions  applicable  only  to  the 
right  triangle.  Actual  test  shows  that  such  rudimentary  methods 
are  wasteful  of  time  and  actually  confirm  the  student  in  narrow- 
ness of  view  and  in  lack  of  scientific  imagination.  For  that  reason, 
the  definitions,  theorems  and  addition  formulas  of  trigonometry 
are  kept  as  general  as  practicable  and  the  formulas  are  given 
general  demonstrations. 

The  possibiUties  and  responsibiUties  of  character  building  in  the 
department  of  mathematics  are  kept  constantly  in  mind.  It  is 
accepted  as  fundamental  that  a  modern  working  course  in  mathe- 
matics should  emphasize  proper  habits  of  work  as  well  as  proper 
methods  of  thought;  that  neatness,  system,  and  orderly  habits 
have  a  high  value  to  all  students  of  the  sciences,  and  that  a  text- 
book should  help  the  teacher  in  every  known  way  to  develop  these 
in  the  student.  ' 

The  present  work  is  a  revision  and  rewriting  of  a  preliminary 
form  which  has  been  in  use  for  three  years  at  the  University  of 
Wisconsin.  During  this  time  the  writer  has  had  frequent  and 
valuable  assistance  from  the  instructional  force  of  the  department 


PREFACE  IX 

of  mathematics  in  the  revision  and  betterment  of  the  text.  Ac- 
knowledgments are  due  especially  to  Professors  Burgess,  Dresden, 
Hart  and  Wolfif  and  to  Instructors  Fry,  Nyberg  and  Taylor. 
Professor  Burgess  has  tested  the  text  in  correspondence  courses, 
and  has  kindly  embraced  that  opportunity  to  aid  very  materially 
in  the  revision.  He  has  been  especially  successful  in  shortening 
graphical  methods  and  in  adapting  them  to  work  on  squared  paper. 
Professor  Wolff  has  read  all  of  the  final  manuscript  and  made 
many  suggestions  based  upon  the  use  of  the  text  in  the  class  room. 
Mr.  Taylor  has  read  all  of  the  proof  and  supphed  the  results  to  the 
exercises. 

Professor  E.  V.  Huntington  of  Harvard  University  has  read  the 
galley  proof  and  has  contributed  many  important  suggestions. 

The  writer  has  avoided  the  introduction  of  new  technical  terms, 
or  terms  used  in  an  unusual  sense.  He  has  taken  the  liberty,  how- 
ever of  naming  the  function  ax",  the  "Power  Function  of  x,"  as  a 
short  name  for  this  important  function  seems  to  be  an  unfortu- 
nate lack — ^a  lack,  which  is  apparently  confined  solely  to  the 
Enghsh  language. 

Chables  S.  Slichtbr. 

University  or  Wisconsin 
July  25,  1914 

Note:  The  results  to  the  exercises  are  issued  aa  a  separate  pamphlet. 


CONTENTS 


Faqb 

Preface .    .      v 

Intbodtjction  ....  .  .    xiii 

Mathematical  Signs  and  Symbols.  ...  .    .       .  xviii 

Chaptbr 

I.  Vakiablbb  and  Functions  op  Vabiablbs  ...       .  1 
II.  Rectanqtjlak   Cookdinatbs    and   the    Straight 

Line  ...           ...                         ....  23 

III.  The  Powbe  Function  .                      48 

MiSCELLANBOITS  ExEBCISES.  ....  92 

IV.  The  Circle  and  the  Circular  Functions  ....     97 
V.  The  Ellipse  and  Htpebbola .152 

VI.  Single  and  Simultaneous  Equations 174 

VII.  Permutations,     Combinations;     the     Binomial 

Theorem .  ...  198 

VIII.  Progressions  .    .  ...  213 

Questions  and  Exercises  for  Review,  Chapters 

I  to  VIII 225 

IX.  The  Logarithmic  and  Exponential  Functions  .  234 
X.  Tbigonombtbic  Equations  and  the  Solution  of 

Tbiangles 304 

XI.  Simple  Harmonic  Motion  and  Waves  .    .           .  339 
XII.  Complex  Numbebs 357 

XIII.  Loci 387 

XIV.  The  Conic  Sections     .    .  399 

XV.  Appendix — A    Review    of    Sbcondabt    School 

Algbbba 451 

Mathematical  Tables 474 

Index 491 


INTRODUCTION 


Any  course  in  mathematics  requires  the  frequent  use  of  geo- 
metrical constructions,  and  the  carrying  out  of  analytical  and 
numerical  computations.  In  order  that  this  work  may  be  per- 
formed neatly  and  accurately  it  is  necessary  that  the  student 
have  a  few  simple  instruments,  and  a  supply  of  proper  material 
for  doing  the  work  in  a  systematic  and  orderly  manner.  The 
indispensible  instruments  are  as  follows : 

I.  Instruments.  (1)  Two  4:H  hexagonal  drawing  pencils;  one 
sharpened  to  a  fine  point  for  marking  points  upon  paper  or  for  sketch- 
ing free  hand;  the  other  sharpened  to  a  chisel  point  for  drawing 
straight  lines.  Some  prefer  to  use  a  single  pencil  sharpened  at  both 
ends,  one  end  round  pointed,  the  other  end  chisel  pointed. 

(2)  A  small  drawing  board'  of  soft  wood — 10X12  inches  is  large 
enough. 

(3)  A  small  T-square  same  length  as  the  drawing  board. 

(4)  A  60°  and  a  45°  transparent  triangle.  Five-inch  triangles 
are  large  enough,  although  a  larger  60°  triangle  will  be  found  to  be 
very  convenient. 

(5)  A  protractor  for  laying  off  angles. 

(6)  A  triangular  boxwood  scale,  decimally  divided. 

(7)  A  pair  of  6-inoh  pencil  compasses  for  drawing  circles  and 
arcs  of  circles,  provided  with  medium  hard  lead,  sharpened  to  a 
narrow  chisel  point. 

(8)  A  10-inch  sUde  rule  is  required  for  Chapter  IX,  and  may  be 
used  earlier  at  the  discretion  of  the  instructor. 

II.  Materials.  All  mathematical  work  should  be  done  on  one 
side  of  standard  size  letter  paper,  8^  X  11  inches.  This  is  the 
smallest  sheet  that  permits  proper  arrangement  of  mathematical 
work.     A  good  equipment  will  include: 

(1)  A  notQ  book  cover  to  hold  sheets  of  the  above  named  size  and 

^Drawing  boards  of  this  size  with  T-square  and  two  wood  triangles  are  marketed 
by  the  Milton  Bradly  Co.,  Springfield,  Mass. 

xiii 


XIV  INTRODUCTION 

a  supply  of  manUa  paper  "vertical  file  folders"  for  use  in  submit- 
ting work  for  the  examination  of  the  instructor. 

(2)  A  number  of  different  forms  of  squared  paper  and  computa- 
tion paper  especially  prepared  for  use  with  this  book.  These  sheets 
will  be  described  from  time  to  time  as  needed  in  the  work.  Form 
M2  wiU  be  found  convenient  for  problem  work  and  for  general  cal- 
culation. M2  is  a  copy  of  a  form  used  by  a  number  of  public  utiUty 
and  industrial  corporations.  Colleges  usually  have  their  own  sources 
of  supply  of  squared  paper,  satisfactory  for  use  with  this  book. 

(3)  Miscellaneous  supphes  such  as  thumb  tacks,  erasers,  sandpaper- 
pencil-sharpeners,  etc. 

in.  General  Directions.  All  drawings  should  be  done  in  pencil, 
unless  the  student  has  had  training  in  the  use  of  the  ruling  pen, 
in  which  case  he  may,  if  he  desires,  "ink  in"  a  few  of  the  most 
important  drawings. 

AU  mathematical  work,  such  as  the  solutions  of  problems  and 
exercises,  and  work  in  computation  should  be  done  in  ink.  The 
student  should  acquire  the  habit  of  working  problems  with  pen 
and  ink.  He  will  find  that  this  habit  will  materially  aid  him  in 
repressing  carelessness  and  indifference  and  in  acquiring  neatness 
and  system. 

TO    THE    STUDENT— SUGGESTIONS    ON    THE    STUDY    OF 
MATHEMATICS 

The  following  suggestions  may  assist  the  student  to  acquire  habits 
of  work  essential  to  success  in  the  study  of  mathematics  and  of  the 
other  exact  sciences. 

Successful  intellectual  work  depends  very  largely  upon  the  power 
of  concentration.  Fortxmately  this  power  can  be  acquired  and  culti- 
vated'. The  student  should  study  away  from  interruption  and  then 
must  not  permit  his  work  to  become  interrupted  by  himseU  or  by 
others.  By  holding  his  attention  upon  his  work  and  by  keeping  his 
mind  from  wandering  to  extraneous  matters,  the  student  will  cultivate 
a  fundamental  habit  that  will  tend  to  assure  his  success  both  in  and 
out  of  college. 

In  a  course  in  mathematics  a  student  (1)  studies  a  textbook  and 
(2)  works  exercises  and  problems.  An  assigrmient  for  a  given  day 
may  therefore  consist  of  the  study  of  mathematical  principles  and 
theory  (such  as  theorems,  definitions,  and  explanations  of  processes), 
or  it  may  consist  of  the  working  out  of  exercises  and  problems,  or, 
as  is  usually  the  case,  it  may  consist  of  the  theory  and  principles  of 


INTRODUCTION  xv 

processes,  together  with  an  assignment  of  exercises  illustrative  of  the 
theory. 

1.  THE  STUDY  OF  THE  TEXTBOOK 

Studying  a  mathematical  textbook  involves  much  more  than 
the  mere  reading  of  the  statements  of  principles  and  of  the  explanation 
of  processes.  The  student  must  usually  read  the  assigned  paragraphs 
several  times  and  must  frequently  turn  back  and  re-read  portions  of 
the  text  included  in  previous  lessons.  In  this  manner  the  various 
points  in  the  reasoning  or  explanations  can  be  thought  over,  and  the 
habit  of  asking  self-put  questions  about  the  work  can  be  acquired. 

First  of  all,  in  preparing  a  lesson,  try  to  find  out  what  'it  is  about — 
what  its  purpose  is.  Try  to  decide  how  you  yourself  would  go  about 
the  aocompUshment  of  the  task  and,  if  possible,  make  an  independent 
attempt  of  your  own.  The  more  consideration  you  give  to  such  an 
attempt,  the  greater  scientific  power  you  will  gain. 

In  particular: 

(A)  The  student  should  remember  that  the  words  in  science  have 
exact  meanings  and,  of  course,  these  meanings  must  be  known  to  the 
student.  In  studying  mathematics  the  student  should  acquire 
and  use  the  language  oi  mathematics.  For  example,  he  should  not 
say  '  equation"  when  he  means  "expression."  Indeed,  he  should  go 
farther  than  this.  He  should  make  a  conscious  effort  to  use  abso- 
lutely correct  English,  not  only  in  written  work  but  in  oral  work 
as  well. 

(B)  While  studying  the  text,  work  out  theorems  or  illustrative 
examples  with  pen  and  ink.  Do  not  rely  upon  a  mere  reading — 
even  repeated  readings — of  a  new  piece  of  reasoning  or  of  the  explana- 
tion of  a  new  process. 

(C)  Bead  over  all  of  the  lesson  assigned  in  the  text  a  last  time  after 
working  the  assigned  exercises.  The  text  will  probably  have  a  new 
meaning  after  working  out  the  special  cases  in  the  exercises.  This 
habit  will  give  a  meaning  to  the  words,  "Learn  by  doing." 

(D)  Finally,  make  a  mental  simimary  of  each  lesson. 

(E)  Review  often. 

2.  THE  WORKING  OF  EXERCISES 

(F)  Read  each  exercise  or  problem  carefully  and  plan  a  method 
of  attack  in  advance  in  order  to  facilitate  arrangements  of  equations 
and  computations  and  the  drawing  of  figures. 

(G)  Look  at  your  result  and  see  if  it  is  a  reasonable  one. 


XVI  INTRODUCTION 

(H)  Check  result. 

(I)  Indicate  the  results  by  a  distinguishing  mark,  or  summarize 
in  logical  qrder. 

(J)  The  figures  and  diagrams  should  have  sufficient  lettering, 
titles,  etc.,  to  make  them  self-explanatory.  The  units  of  measure 
used  should,  of  course,  be  clearly  indicated. 

(K)  Do  all  work  neatly  the  first  time  and  (except  drawings) 
invariably  in  ink.  Try  to  have  the  first  draft  sufficiently  neat  in 
appearance  and  arrangement  to  hand  in  to  your  instructor. 

(L)  After  the  first  draft  has  been  finished,  read  it  over  carefully 
to  see  where  it  may  be  unproved  in  method  or  arrangement  and 
think  about  the  processes  you  have  used.  If  small  changes  only 
are  needed  to  effect  the  desired  improvement,  make  them  by  drawing 
lines  through  the  portions  to  be  changed  and  by  making  neat  inser- 
tions.    If  considerable  changes  are  necessary,  do  the  work  over. 

The  study  and  improvement  of  the  work  will  prove  to  be  of  fully 
as  much  importance  to  you  as  the  doing  of  the  work  itself. 

(M)  See  to  it  that  each  piece  of  work  or  exercise  is  complete.  On 
any  piece  of  written  work  the  nature  of  the  problem  should  be  clearly 
and  briefly  stated.  The  student  should  learn  to  think  of  each  piece 
of  work  as  a  thing  that  is  in  itself  worth  while.  Hence  each  detail 
should  be  attended  to  before  the  work  is  submitted  to  the  instructor. 
See  that  sufficient  explanation  is  given  and  that  the  numbers  and 
magnitudes  are  adequately  named  and  labelled. 

TO  THE  INSTRUCTOR 

The  instructor  cannot  insist  too  emphatically  upon  the  require- 
ment that  all  mathematical  work  done  by  the  student — ^whether 
preliininary  work,  numerical  scratch  work,  or  any  other  kind  (except 
drawings) — shall  be  carried  out  with  pen  and  ink  upon  paper  of 
suitable  size.  This  should,  of  course,  include  all  work  done  at  home, 
irrespective  of  whether  it  is  to  be  submitted  to  the  instructor  or  not. 
The  "psychological  effect"  of  this  requirement  will  be  found  to  entrain 
much  more  than  the  acquirement  of  mere  technique.  If  properly 
insisted  upon,  orderly  and  systematic  habits  of  work  will  lead  to 
orderly  and  systematic  habits  of  thought.  The  final  results  will  be 
very  gratifying  to  those  who  sufficiently  persist  in  this  requirement. 

At  institutions  whose  requirements  for  admisstion  include  more 
than  one  and  one-half  units  of  preparatory  algebra,  nearly  all  of 
Chapters  VI,  VII,  and  VIII  may  be  omitted  from  the  course. 

An  asterisk  attached  to  a  section  number  indicates  that  the  section 


INTRODUCTION  xvii 

may  he  omitted.  These  sections  will  frequently  be  found  useful  in 
forming  the  basis  of  discussion  by  the  instructor. 

The  usual  one  and  one-half  year  of  secondary  school  Algebra, 
including  the  solution  of  quadratic  equations  and  a  knowledge  of 
fractional  and  negative  exponents,  is  required  for  the  work  of  this 
course.  In  the  appendix  (Chapter  XV)  vrill  be  found  material  for  a 
brief  review  of  factoring,  qitadratics,  and  exponents,  upon  which  a  week 
or  ten  days  should  be  spent  before  beginning  the  regular  work  in  this 
text. 

This  review  chapter  is  placed  last  because  the  amount  of  material 
in  it  is  greater  than  need  be  taken  in  all  cases  and  also  because  college 
students  do  not  like  to  be  confronted  on  the  &st  page  of  a  scientific 
text-book  with  elementary  work  of  high  school  grades. 


GREEK  ALPHABET 


Capitals 

Lower 
case 

Names 

Capitals 

Lower 
case 

Names 

A 

a 

Alpha 

N 

V 

Nu 

B 

P 

Beta 

S 

i 

Xi 

r 

y 

Gamma 

0 

o 

Omicron 

A 

s 

Delta 

II 

IT 

Pi 

E 

e 

Epsilon 

p 

9 

Rho 

Z 

f 

Zeta 

s 

a 

Sigma 

,     H 

V 

Eta 

T 

T 

Tau 

e 

e 

Theta 

T 

V 

Upsilon 

I 

L 

Iota 

* 

<t> 

Phi 

K 

K 

Kappa 

X 

X 

Chi 

A 

X 

Lambda 

* 

i' 

Psi 

M 

/< 

Mu 

Q 

U> 

Omega 

MATHEMATICAL  SIGNS  AND  SYMBOLS 


read 

= 

read 

5^ 

read 

= 

read 

=F 

read 

> 

read 

< 

read 

1^ 

read 

(a,b) 

read 

|n 

read 

n! 

read 

limit  r,,  ,' 

read 

a;  =  oo 

read 

\a\ 

read 

log„a; 

read 

Iga; 

read 

In  a; 

read 

and  so  on. 

is  identical  vnth. 

is  not  equal  to. 

approaches. 

is  approximately  equal  to. 

is  greater  than. 

is  less  than. 

is  greater  than  or  equal  to. 

point  whose  coordinates  are  a  and  b. 

factorial  n. 

factorial  w  or  n  admiration. 

limit  of  fix)  as  x  approaches  a. 

X  becomes  infinite, 
absolute  value  of  a. 
logarithm  of  x  to  the  base  a. 
common  logarithm  of  x. 
natural  logarithm  of  x. 


Sw„ 


read       summation  from  n  =  1  io  n  =  r  of  u„ 


ELEMENTARY 
MATHEMATICAL  ANALYSIS 

CHAPTER  I 
VARIABLES  AND  FUNCTIONS  OF  VARIABLES 

1.  Scales.  Select  a  series  of  points  along  any  curve  and  mark 
the  points  of  division  with  the  numbers  of  any  sequence.'  The 
result  of  such  a  construction  is  called  a  scale.  Thus  in  Fig.  1 
the  points  along  the  curve  OA  have  been  selected  and  marked  in 
order  with  the  numbers  of  the  sequence: 

Oj  4j  2>  1>  25,  3,  5,  7,  8 


A  non-uniform  scale. 


Thus  primitive  man  might  have  made  notches  along  a  twig 
and  then  made  use  of  it  in  making  certain  measurements  of 
interest  to  him.  If  such  a  scale  were  to  become  generally  used  by 
others,  it  would  be  desirable  to  make  many  copies  of  the  original 
scale.  It  would,  therefore,  be  necessary  to  use  a  twig  whose  shape 
could  be  readily  duplicated;  such,  for  example,  as  a  straight 
stick;  and  it  would  also  be  necessary  to  attach  the  same  symbols 
invariably  to  the  same  divisions. 

Certain  advantages  are  gained  (often  at  the  expense  of  others, 
however)  if  the  distances  between  consecutive  points  of  division 
are  kept  the  same;  that  is,  when  the  intervals  are  laid  off  by  repe- 
tition of  the  same  selected  distance.  When  this  is  done,  the  scale 
is  called  a  uniform  scale.    Primitive  man  might  have  selected  for 

1 A  sequence  of  numbers  here  means  a  set  of  numbers  arranged  in  order  of 
magnitude. 

1 


2  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§1 

such  uniform  distance  the  length  of  his  foot,  or  sandal,  the  breadth 
of  his  hand,  the  distance  from  elbow  to  the  end  of  the  middle 
finger  (the  cubit),  the  length  of  a  step  in  pacing  (the  yard),  the 
amount  he  can  stretch  with  both  arms  extended  (the  fathom), 
etc. 


Fig.  2. — An  ammeter  scale. 


We  are  familiar  with  many  scales,  such  as  those  seen  on  a 
yardstick,  the  dial  of  a  clock,  a  thermometer,  a  sun-dial,  a  steam- 
gage,  an  ammeter  or  voltmeter,  the  arm  of  a  store-keeper's 
scales,  etc.    The  scales  on  a  clock,  a  yardstick,  or  a  steel  tape  are 

uniform.  Those  on  a  sun-dial, 
on  some  ammeters  or  on  a  good 
thermometer,  are  not  uniform. 
One  of  the  most  important 
advantages  of  a  uniform  scale 
is  the  fact  that  the  place  of 
beginning,  or  zero,  maybe  taken 
at  any  one  of  the  points  of  divi- 
sion. This  is  not  true  of  a  non- 
uniform scale.  If  a  sun-dial  is  not  properly  oriented,  it  is  useless. 
If  the  needle  of  an  ammeter  be  bent  the  instrument  cannot  be  used. 
It  is  always  necessary  in  using  such  an  instrument  to  know  that 
the  zero  is  correct.  If,  however,  a  yarfistick  or  a  steel  tape  be 
broken,  it  may  stUl  be  used  for  measuring  lengths.     The  student 


P.M 


A.M. 


Sun-dial  scale. 


§2]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES  3 

may  think  of  many  other  advantages  gajned  in  using  a  uniform 
scale. 

2.  Formal  Definition  of  a  Scale.  If  points  be  selected  in  order 
along  any  curve  corresponding,  one  to  one,  to  the  numbers  of 
any  sequence,  the  curve,  with  its  divisions,  is  called  a  scale. 

The  notion  of  one  to  one  correspondence,  included  in  this 
definition,  is  frequently  used  in  mathematics. 

Ii  II  I  h  I  II  h  I  I  I  I  111  il  I  M  I  I  II  II  h  I  I  I  h  I  I  I  h  I  I  I  I  II  I  I  I 

0  1.2  3  1  5 

Fig.  4. — A  uniform  arithmetical  scale. 

In  mathematics  we  frequently  speak  of  the  arithmetical  scale 
and  of  the  algebraic  scale.  The  arithmetical  scale  corresponds 
to  the  numbers  of  the  sequence 

0,  1,  2,  3,  4,  5,  .       . 

and  such  intermediate  numbers  as  may  be  desired.    It  is  usually 
represented  by  a  uniform  scale  as  in  Fig.  4.    The  algebraic 
scale  corresponds  to  the  numbers  of  the  sequence 
.    .    .    -6, -5, -4, -3,-2,-1,0,+!, +2, +3, +4, +5,  .    .    . 
and  such  intermediate  numbers  as  may  be  desired.    It  is  usually 

1  I   I  I   I   I   I   I    1  I   I  I   I   I    I  I   11  I   I  I   M   I   I   I   I   I   I  I  I   I   I   I   I   I  I   I   I   I    I  I   II   I   I   II  I   I  1 

-B       -4        -S       -2       -1  0       +1       +2       +3       +4      +5 

Fig.  5. — A  uniform  algebraic  scale. 

represented  by  a  uniform  scale  as  in  Fig.  5.  The  arithmetical 
scale  begins  at  0  and  extends  indefinitely  in  one  direction.  The 
algebraic  scale  has  no  point  of  beginning;  the  zero  is  placed  at  any 
desired  point  and  the  positive  and  negative  numbers  are  then 
attached  to  the  divisions  to  the  right  and  the  left,  respectively, 
of  the  zero  so  selected.  The  algebraic  scale  extends  indefinitely 
in  both  directions. 

Exercises 

1.  On  a  uniform  algebraic  scale,  how  far  is  the  point  marked  5  from 
the  point  marked  7?  How  far  is  the  point  marked  6  from  the  point 
marked  10.5?  How  far  is  th'e  point  marked  —10.8  from  the  point 
marked  13.6? 


4  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§3 

2.  Show  that  the  distance  between  two  points  selected  anywhere  on 
the  uniform  algebraic  scale  is  always  found  by  subtraction. 

3.  What  points  of  the  uniform  algebraic  scale  are  distant  5  from  the 
point  3  of  that  scale?  What  point  of  the  uniform  arithmetical  scale 
is  distant  5  from  the  point  3  of  that  scale? 

4. -If  two  algebraic  scales  intersect  at  right  angles,  the  common 
point  being  the  zero  of  both  scales,  explain  how  to  find  the  distance 
from  any  point  of  one  scale  to  any  point  of  the  other  scale. 

3.  Two  Uniform  Scales  in  Juxtaposition  or  Double  Scales. 

The  relation  between  two  magnitudes  or  quantities,  or  between 
two  numbers,  may  be  shown  conveniently  by  placing  two  scales 
side  by  side.  Thus  the  relation  between  the  number  of  centi- 
meters and  the  number  of  inches  in  any  length  may  be  shown 
by  placing  a  centimeter  scale  and  a  foot-rule  side  by  side  with 
their  zeros  coinciding  as  in  Fig.  6.  From  this  figure  it  is  seen 
that  1  inch  corresponds  to  2.6  centimeters;  3.3  inches  correspond 
to  8.44  centimeters,  4.6  inches  corresponds  to  11.76  centimeters; 
that  5  centimeters  correspond  to  1.97  inches,  8.5  centimeters 
corresponds  to  3.32  inches,  etc. 

A  thermometer  is  frequently  seen  bearing  both  Fahrenheit  and 
the  centigrade  scales.  See  Fig.  7.  It  is  obvious  that  the  double 
scale  of  such  a  thermometer  may  be  used  (within  the  limits  of  its 
range)  for  converting  any  temperature  reading  Fahrenheit  into 
the  corresponding  centigrade  equivalent  or  vice  versa.  From 
Fig.  7  it  is  seen  that  72°F.  corresponds  to  22.2°C.,  212°F.  to 
100°C.,  32°F.  to  zero  degrees  centigrade;  that  21°C.  corresponds 
to  69.8°F.,  72''C.  to  161.6°F. 

The  construction  of  scales  of  the  kind  considered  above 
may  be  made  to  depend  upon  the  following  problem  in  elementary 
geometry :     To  divide  a  given  line  into  a  given  number  of  equal  parts. 

Illustration.  In  Fig.  9  is  given  a  double  scale  OA-OB  showing  the 
correspondence  between  speed  expressed  in  miles  per  hour  and  speed 
expressed  in  feet  per  second.  The  student  will  reproduce  neatly  and 
accurately  the  drawing,  on  a  larger  scale,  in  accordance  with  the 
directions  given  below, 

A  mUe  contains  5280  feet,  an  hour  contains  3600  seconds.  Hence, 
one  mile  per  hour  equals  |f-§-J  or  -f-|  feet  per  second.  Therefore, 
if  two  uniform  scales.  Fig.  9,  one  rejJresenting  speed  expressed  in 
feet  per  second,  and  the  other  representing  speed  expressed  in  miles 


§3]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES 


-r-S 


_  — CO  o 


-pas 

o  « 
I 

a        «3 

6 

M 


o o  c3 


&0 


I 


-         ag 

o  V 

■     g  g 
=•  »•§ 


•a  ^ I 

S  "»  _ 

O 


I-         _ 


5?  fe 


o 


o 


00 


f=( 


6  ELEMENTARY  MATHEMATICAL  ANALYSIS         [§3 


20- 


B- 


18- 


11- 


i,;i2- 


1^ 


6- 


/o. 


Fia.  9. — Method  of  constraction  of  double  scale  showing  relation 
between  "miles  per  hour"  and  "feet  per  second." 


§3]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES  7 

per  hour,  are  constructed  with  their  zeros  coinciding  and  with  the 
point  marked  22  of  the  first  coinciding  with  the  point  marked  15  of  the 
second,  the  double  scale  may  be  used  for  converting  speed  expressed 
as  miles  per  hour  into  speed  expressed  as  feet  per  second,  or  vice  versa. 

Lay  off  with  a  scale  a  line  OA  11  inches  long.  Divide  this  line 
into  22  equal  parts,  and  subdivide  each  division  into  6  equal  parts. 
Mark  these  divisions  and  subdivisions  as  indicated  in  Fig.  9.  Draw 
the  line  OK,  making  the  angle  KOB  about  30°.  With  a  pair  of  bow 
dividers  or  with  a  scale  lay  off  on  OK  15  equal  divisions,  about  f  of 
an  inch  each.  Let  the  last  point  of  division  be  marked  C.  OC  is 
then  divided  into  15  equal  parts.  Draw  CA.  With  a  pair  of  triangles 
draw  lines  through  the  points  Ci,  Ci,  Cz,  .     Cn  parallel  to  CA, 

intersecting  the  line  OA  in  the  points  marked  15,  14,  13,  1, 

respectively.     Why  is  OB  a  uniform  scale  divided  into  15  equal  parts? 

Mark  the  scales  OA  and  OB  in  red  ink  with  a  new  set  of  numbers 
so  that  the  double  scale  may  also  be  used  for  converting  speeds  if 
the  readings  fall  between  15  and  30  feet  per  second  instead  of  between 
0  and  15. 

From  the  double  scale  just  constructed,  find  the  speeds  expressed 
as  miles  per  hour  corresponding  to  speeds  of  2,  4,  5,  11,  14,  20,  and  25 
feet  per  second. 

The  lengths  selected  to  represent  the  various  units  in  any  dia- 
gram are,  of  course,  arbitrary.  As,  however,  the  student  is 
expected  to  prepare  the  various  constructions  and  diagrams 
required  for  the  exercises  in  this  book  on  paper  of  standard 
letter  size  (that  is,  8 J  by  11  inches),  the  units  selected  should 
be  such  as  to  permit  a  convenient  and  practical  construction 
upon  sheets  of  that  size. 

Exercises 

The  student  is  expected  to  carry  out  the  actual  construction  of  only 
one  of  the  double  scales  described  in  the  following  exercises. 

1.  Draw  a  double  scale  showing  the  relation  between  pressure 
expressed  as  inches  of  mercury  and  as  feet  of  water,  knowing  that 
the  density  of  mercury  is  13.6  times  that  of  water. 

These  are  two  of  the  common  ways  of  expressing  pressure.  Water 
pressure  at  water  power  plants,  and  often  for  city  water  service,  is 
expressed  in  terms  of  head  in  feet.  Barometric  pressure,  and  the 
vacuum  in  the  suction  pipe  of  a  pump  and  in  the  exhaust  of  a  con- 
densing steam  engine  are  expressed  in  inches  of  mercury.  The 
approximate  relations  between  these  units,  i.e.,   1  atmosphere  =  30 


8  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§3 

inches  of  mercury  =  32  feet  of  water'  =  15  pounds  per  square  inch, 
are  known  to  every  student  of  elementary  physics.  To  obtain,  in 
terms  of  feet  of  water,  the  pressure  equivalent  of  a;  feet  of  mercury, 
multiply  X  by  13.6,  the  specific  gravity  of  mercury.  This  product 
divided  by  12,  or  1.13a;,  gives  the  number  of  feet  of  water  corre- 
sponding to  X  inches  of  mercury. 

If  we  let  the  scale  of  inches  of  mercurj'  range  from  0  to  10,  then  the 
scale  of  feet  of  water  must  range  from  0  to  11.3.  Hence  draw  a  line 
OA  10  inches  long  divided  into  inches  and  tenths  to  represent  inches 
of  mercury.  Draw  any  line  OC  through  0  and  lay  off  on  it  11.3  uni- 
form intervals  (uich  intervals  will  be  satisfactory).  Connect  the 
end  division  on  OA  with  the  end  division  on  OC  by  a  line  AC.  Then 
from  1,  2,  3,  inches  on  OC  draw  parallels  to  AC,  thus  forming 

adjacent  to  OA  the  scale  of  equivalent  feet  of  water.  Each  of  these 
intervals  can  then  be  subdivided  into  10  equal  parts  corresponding 
to  tenths  of  feet  of  water. 

2.  Draw  a  double  scale  showing  pressure  expressed  as  feet,  of  water, 
and  as  pounds  per  square  inch,  knowing  that  one  cubic  foot  of  water 
weighs  62.5  pounds. 

The  weight  of  one  cubic  foot  of  water,  62.5  pounds,  divided  by  144, 
the  number  of  square  inches  on  one  face  of  a  cubic  foot,  gives  0.434 
pounds  per  square  inch  as  the  equivalent  of  one  foot  of  water 
pressure. 

One  pound  per  square  inch  is  equivalent,  therefore,  to  1/0.434  or 
2.30  feet  of  water  pressure.  If  we  let  the  scale  of  pounds  range  from 
0  to  10,  we  may  select  1  inch  as  the  equivalent  of  1  pound  per 
square  inch,  and  divide  the  scale  OA  into  inches  and  tenths  to  repre- 
sent this  magnitude.  Draw  OC  through  0,  and  lay  off  23  uniform 
intervals  on  OC,  1/2  inch  being  a  convenient  length  for  each  of  these 
parts.  Connect  the  end  division  of  OC  with  A  and  through  all 
points  of  division  of  OC  draw  lines  parallel  to  CA.  The  range  may 
be  extended  to  any  amount  desired  by  annexing  ciphers  to  the 
numbers  attached  to  the  two  scales. 

Extending  the  range  by  annexing  ciphers  to  the  attached  numbers 
is  obviously  practicable  so  long  as  the  various  intervals  or  units  are 
decimally  subdivided.  The-  method  is  impracticable  for  scales  that 
are  not  decimally  subdivided,  such  as  shilUngs  and  pence,  degrees  and 
minutes,  feet  and  inches,  etc. 

3.  Draw  a  double  scale  showing  the  relations  between  cubic  feet, 
and  gallons.  One  gallon  equals  231  cubic  inches,  but  use  the 
approximate  relation,   1  cubic  foot  equals  71  gallons.     Divide  the 


§4]        VABIABLES  AND  FUNCTIONS  OF  VARIABLES  9 

scale  of  cubic  feet  into  tenths,  the  scale  of  gallons  into  fourths  to 
correspond  to  quarts. 

It  is  obvious  that  it  is  always  necessary  first  to  select  the  range 
of  the  various  scales,  but  it  is  quite  as  well  in  this  case  to  show  the 
equivalents  for  1  cubic  foot  only,  as  numbers  on  the  various  scales 
can  be  multiplied  by  10,  100,  or  1000,  etc.,  to  show  the  equivalents  for 
larger  amounts. 

Select  10  inches  =  1  cubic  foot  for  the  scale  (OA)  of  cubic  feet. 
Draw  the  line  OC.  On  OC  lay  off  71  equal  parts  (say,  7^  inches). 
Connect  the  end  division  with  A  and  draw  the  parallel  lines  exactly 
as  with  previous  examples.  The  intervals  of  the  scale  of  gallons  can 
then  be  subdivided  into  the  four  equal  parts  to  show  quarts. 

4.  Draw  a  double  scale  showing  the  relation  between  cubic  feet 
and  liters.     One  cubic  foot  equals  28|  liters. 

5.  If  a  double  scale  be  drawn  on  a  deformable  body,  as,  for  example, 
on  a  rubber  band,  would  the  double  scale  still  represent  true  relations 
when  the  rubber  band  is  stretched?  What  if  the  stretching  were 
not  uniform? 

6.  From  Fig.  9,  find  the  number  of  miles  per  hour  corresponding 
to  0,  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  and  15  feet  per  second. 
Place  the  results  in  tabular  form,  i.e.,  in  the  first  of  two  adjacent 
vertical  columns  place  the  numbers  0,  1,  2,  .  .  .15;  opposite  these 
numbers  place  in  the  second  vertical  column  the  corresponding 
numbers  representing  speed  as  mUes  per  hour.  Give  the  first  vertical 
column  the  heading  "Speed-ft./sec,"  and  the  second  column  the 
heading  "Speed-mi./hr."  Aa  the  speed  changes  from  1  foot  per 
second  to  2  feet  per  second,  the  speed  changes  by  what  amount  in 
miles  per  hour?  As  the  speed  changes  from  3  feet  per  second  to  4 
feet  per  second,  the  speed  in  miles  per  hour  changes  by  what  amount? 
The  change  in  speed  as  miles  per  hour  is  how  many  times  the  change  in 
the  speed  as  feet  per  second? 

4.  A  Non-uniform  Scale  in  Juxtaposition  with  a  Uniform  scale. 

Each  scale  of  the  double  scales  constructed  in  the  preceding 
section  were  uniform  scales.  The  construction  of  a  double  scale 
of  this  kind  was  possible  because  the  change  in  the  number  of  units 
of  one  magnitude  represented  was  directly  proportional  to  the 
corresponding  change  in  the  number  of  units  of  the  other 
magnitude.  It  will,  however,  be  sometimes  desirable  to  construct 
double  scales  in  which  this  proportionality  does  not  exist.  For 
example,  if  a  double  scale  were  to  be  constructed  showing  the 


10  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§5 

relation  between  the  radius  and  the  area  of  a  circle,  the  preceding 
construction  could  not  be  used,  since  the  change  in  area  is  pro- 
portional to  the  change  in  the  square  of  the  radius  and  not 
to  the  change  in  the  radius.  In  this  case  both  scales  cannot 
be  uniform.  Figure  10  is  a  double  scale  representing  the  relation 
between  the  radius  and  the  area  of  a  circle.  The  area  is  repre- 
sented by  the  points  on  the  uniform  scale,  the  radius  by  the  points 
on  the  non-uniform  scale.  The  relation  is  A  =  irr^  where  r  is 
the  radius  in  feet  and  A  is  the  area  in  square  feet. 


KadluB  ol  circle 
4  5  6 


iMiliMil[iiil||MliiiilMiJ|iMl|iii[iiiiliii-rlm^ilyi|li|l|ll||-|l|iyl^il|li^^ 


Illlllllll    llllllllllu 


Area  of  Circle 

Fig.  10. — Double  scale  showing  the  relation  between  the  area  of  a 
circle  and  its  radius. 

5.  Functions.  The  relation  between  two  magnitudes  expressed 
graphically  by  two  scales  drawn  in  juxtaposition,  as  above,  may 
sometimes  be  expressed  by  means  of  an  equation.  Thus,  F, 
the  number  representing  the  degrees  Fahrenheit  in  a  temperature 
reading,  and  C,  the  number  representing  the  degrees  centigrade 
of  the  same  temperature,  are  connected  by  the  equation 

F  =  iC  +  32.  (1) 

Again  y,  the  number  representing  speed  measured  as  miles  per 
hour,  and  x,  the  number  representing  speed  measured  as  feet 
per  second,  are  connected  by  the  equation 

y  =  iU.  (2) 

Again  u,  the  number  representing  pressure  measured  as  feet  of 
water,  and  v,  the  number  representing  the  same  pressure  measured 
as  pounds  per  square  inch,  are  connected  by  the  equation 

u  =  U^j^-  (3) 

Again  A,  the  number  representing  area  of  a  circle  measured  as 
square  feet,  and  r,  representing  the  radius  measured  as  feet,  are 
connected  by  the  relation  A  =  irr^.  (4) 

Note.  The  letters  F,  C,  x,  y,  u,  v,  in  the  above  equations  stand 
for  numbers; -to  make  this  emphatic  we  sometimes  speak  of  them  as 


§5]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES        11 

pure  or  abstract  numbers.  These  numbers  are  thought  of  as  arising 
from  the  measurement  of  a  magnitude  or  quantity  by  the  appUoation 
of  a  suitable  unit  of  measure.  Thus  from  the  magnitude  or  quantity 
of  water,  12  gallons,  arises,  by  use  of  the  unit  of  measure  the  gallon, 
the  abstract  number  12. 

Algebraic  equations  express  the  relation  between  numbers,  and  it  is 
understood  that  the  letters  used  in  algebra  stand  for  numbers  and 
not  for  quantities  or  magnitudes. 

Quantity  or  Magnitude  is  an  answer  to  the  question:  "How 
much?"    Number  is  an  answer  to  the  question:  "How  many?" 

An  interesting  relation  is  given  by  the  scales  in  Fig.  8.  This 
diagram  shows  the  fee  charged  for  money  orders  of  various 
amounts.  The  amount  of  the  order  may  first  be  found  on  the 
upper  scale  and  then  the  amount  of  the  fee  may  be  read  from  the 
lower  scale.  The  relation  here  exhibited  is  quite  different  from 
those  previously  given.  For  example,  note  that  as  the  amount  of 
the  order  changes  from  $50.01  to  $60  the  fee  does  not  change,  but 
remains  fixed  at  20  cents.  Then  as  the  amount  of  the  order 
changes  from  $60.00  to  $60.01,  the  fee  changes  abruptly  from 
20  cents  to  25  cents.  For  an  order  of  any  amount  there  is  a  cor- 
responding fee,  but  for  each  fee  there  corresponds  not  an  order  of 
a  single  value,  but  orders  of  a  considerable  range  in  value.  This  is 
quite  different  from  the  cases  presented  in  Fig.  7.  There  for  each 
reading  Fahrenheit  corresponds  a  certain  reading  centigrade, 
or  vice  versa,  and  for  any  change,  however  small,  in  one  of  the 
temperature  readings  a  change,  also  small,  takes  place  in  the 
other  reading.  For  this  reason  the  latter  number  is  said  to  be 
continuous. 

The  relation  between  the  temperature  scales  has  been  expressed 
by  an  algebraic  equation.  The  relation  between  the  value  of  a 
money  order  and  the  corresponding  fee  cannot  be  expressed  by  a 
similar  equation.  If  we  had  given  only  a  short  piece  of  the  centi- 
grade-Fahrenheit double  scale,  we  could,  nevertheless,  produce  it 
indefinitely  in  both  directions,  and  hence  find  the  corresponding 
readings  for  all  desired  temperatures.  But  by  knowing  the  fees 
for  a  certain  range  of  money  orders  we  cannot  determine  the 
fees  for  other  amounts.  In  both  of  these  cases,  however,  we 
express  the  fact  of  dependence  of  one  number  upon  another 


12  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§6 

number  by  sajnng  that  the  first  number  is  a  function  of  the  second 
number. 

6.  Definition.  Any  number,  u,  is  said  to  be  a  function  of 
another  number,  t,  if,  when  the  value  of  t  is  given,  the  value  of  u 
is  determined.  The  number  t  is  called  the  argument  of  the 
function  u. 

Illustrations.  The  length  of  a  rod  is  a  function  of  its  tempera- 
ture. The  area  of  a  square  is  a  function  of  the  length  of  a  side. 
The  area  of  a  circle  is  a  function  of  its  radius.  The  square  root 
of  a  number  is  a  function  of  the  number.  The  strength  of  an 
iron  rod  is  a  function  of  its  diameter.  The  pressure  in  the  ocean 
is  a  function  of  the  depth  below  the  surface.  The  price  of  a 
railroad  ticket  is  a  function  of  the  distance  to  be  travelled. 
The  temperature  Fahrenheit  is  a  function  of  the  temperature 
centigrade. 

It  is  obvious  that  any  mathematical  expression  is,  by  the  above 
definition,  a  function  of  the  letter  or  letters  that  occur  in  it. 
Thus,  in  the  equations 

u  =  t^  +  it  +  l 

_   t  -  1 
"  ~  2(  +  2 

u  =  Vt  +  4:  +  P  -\ 

u  is  in  each  case  a  function  of  t. 

Goods  sent  by  freight  are  classified  into  first,  second,  third, 
fourth,  and  fifth  classes.  The  amount  of  freight  on  a  package  is 
a  function  of  its  class.  It  is  also  a  function  of  its  weight.  It  is 
also  a  function  of  the  distance  carried.  Only  the  second  of  these 
functional  relations  just  named  can  readily  be  expressed  by  an 
algebraic  equation.  It  is  possible,  however,  to  express  all  three 
graphically  by  means  of  parallel  scales.  The  definition  of  the 
function  is  given  (for  any  particular  railroad)  by  the  complete 
freight  tariff  book  of  the  railroad. 

The  fee  charged  for  a  money  order  is  a  function  of  the  amount  of 
the  order.  The  functional  relation  has  been  expressed  graphically 
in  Fig.  8.  Note  that  for  orders  of  certain  amounts,  namely, 
$2i  $5,  $10,  $20,  $30,  $40,  $50,  $60,  $75,  the  function  is  not  de- 
fined.   The  graph  alone  cannot  define  the  function  at  these  values, 


§6]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES         13 

as  one  cannot  know  whether  the  higher,  the  lower,  or  an  inter- 
mediate fee  should  be  demanded.  One  can,  however,  define 
the  function  for  these  values  by  the  supplementary  statement 
(for  example):  "For  the  critical  amounts,  always  charge  the 
higher  fee."  As  a  matter  of  fact,  however,  the  lower  fee  is  always 
charged. 

A  function  having  sudden  jumps  like  the  one  just  considered, 
is  said  to  be  discontinuous. 

Illustration  1.  One  side  of  a  rectangle  is  2  centimeters.  The  other 
side  is  (x  +  2)  centimeters.  Express  the  area  A  of  the  rectangle  as 
a  function  of  x. 

The  area  is  the  product  of  the  breadth  by  the  length  of  the  rectangle. 
Hence 

A  =2(x  +  2)  =2x  +  4,  (l; 

which  is  the  function  of  x  sought. 

Illustration  2.  The  hypotenuse  of  a  right  triangle  is  10  inches. 
One  side  is  x  inches.  Express  A,  the  area  of  the  triangle,  as  a  function 
of  x. 

Since  the  hypotenuse  squared  equals  the  sum  of  the  squares  of  the 
two  legs,  we  may  write 

102   =  a;2  -f  yi^  (1) 

where  y  stands  for  the  length  in  inches  of  the  second  leg  of  the  triangle. 
But  we  know  that 

A  =  kxy.  (2) 

From  (1)  

y  =  Vl02  -  xS  '  (3) 

Substituting  in  (2),  we  have 

A  =  kxy/im  -x\  (4) 

which  is  the  function  of  x  desired.  , 

Illustration  3.  Express  the  amount  A  of  $1  at  simple  interest  at 
6  per  cent,  for  n  years  as  a  function  of  n. 

The  interest  on  $1  for  n  years  equals  Sy^Tfre.  Hence  the  amount 
(which  is  the  principal  plus  the  interest)  is'  expressed  by 

,''   A    =   1   +  TBTO. 

Exercises 

In  the  following  exercises  the  function  described  can  be  represented 
by  a  mathematical  expression.  The  problem  is  to  set  up  the  expres- 
sion in  each  case. 


14  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§7 

1.  One  side  of  a  rectangle  is  10  feet.  Express  the  area  il  as  a  func- 
tion of  the  other  side  x. 

2.  One  leg  of  a  right  triangle  is  15  feet.  Express  the  area  .A  as  a 
function  of  the  other  leg  x. 

3.  The  base  of  a  triangle  is  12  feet.  Express  the  area  as  a  function 
of  the  altitude  I. 

4.  Express  the  circumference  of  a  circle  as  a  function  (1)  of  its 
radius  r;  (2)  of  its  diameter  d. 

6.  Express  the  diagonal  doia,  square  as  a  function  of  one  side  x, 

6.  One  leg  of  a  right  triangle  is  10.  Express  the  hypotenuse  h  as 
a  function  of  the  other  leg  x. 

7.  A  Ship  B  sails  on  a  course  AB  perpendicular  to  OA.  If  OA  =  30 
mUes,  express  the  distance  of  the  ship  from  0  as  a  function  of  AB. 

8.  A  circle  has  a  radius  10  units.  Express  the  length  of  a  chord 
as  a  function  of  its  distance  from  the  center. 

9.  An  isosceles  triangle  has  two  sides  each  equal  to  15  centimeters, 
and  the  third  side  equal  to  x  centimeters.  Express  the  area  of  the 
triangle  as  a  function  of  x. 

10.  A  right  cone  is  inscribed  in  a  sphere  of  radius  12  inches.  Ex- 
press the  volume  of  the  cone  as  a  function  of  its  altitude  I. 

Hint:  The  distance  from  the  center  of  the  sphere  to  the  base  of 
the  cone  is  (t— 12),  if  I  >12.  The  radius  of  the  base  of  the  cone  is 
Vl2'-(.l-12)'  or  V24J-Z2.     What  if  2  <  12? 

11.  A  right  cone  is  inscribed  in  a  sphere  of  radius  a.  Express  the 
volume  of  the  cone  as  a  function  of  its  altitude  I. 

12.  One  dollar  is  at  compound  interest  for  20  years  at  r  per  cent. 
Express  the  amount  A  as  a  function  of  r. 

7.  Functional  Notation.  The  following  notation  is  used  to  ex- 
press that  one  number  is  a  function  of  another;  thus,  if  u  is  a 
function  of  t  we  write 

Likewise 

y  =  /W 
means  that  y  is  a  function  of  x.    Other  symbols  commonly  used  to 
express  functions  of  x  are : 

Hx),  Xix),  f'(x),  F(x),  etc. 

These  may  be  read  the  "(^-function  of  x,"  the  "Z-function  of  x," 
etc.,  or  more  briefly,  "the  <l>  of  x,"  "the  X  of  x,"  etc. 
Expressing  the  fact  that  temperature  reading  Fahrenheit  (.F)  is 


§8]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES        15 

a  function  of  temperature  reading  centigrade  (C),  we  may  write: 

F=f(.C). 
This  is  made  specific  by  writing 

F  =  iC  +  32. 

Likewise  the  fact  that  the  charge  for  freight  is  a  function  of  class, 
weight,  and  distance,  may  be  written 

r  =  fie,  w,  d). 

To  make  this  functional  symbol  explicit,  might  require  that  we  be 
furnished  with  the  complete  schedule  as  printed  in  the  freight  tariff 
book  of  the  railroad.  The  dependence  of  the  tariff  upon  class  and 
weight  can  usually  be  readily  expressed,  but  the  dependence  upon 
distance  often  contains  arbitrary  elements  that  cause  it  to  vary 
irregularly,  even  on  different  branches  of  the  same  railroad.  A 
complete  specification  of  the  functional  symbol  /  would  be  con- 
sidered given  in  this  case  when  the  tariff  book  of  the  railroad  was  in 
our  hands. 

8.  Variables  and  Constants.  In  elementary  algebra,  a  letter  is 
always  used  to  stand  for  a  number  that  preserves  the  same  value 
in  the  same  problem  or  discussion.  Such  numbers  are  called 
constants.  In  the  discussion  above  we  have  used  letters  to  stand 
for  numbers  that  are  assumed  not  to  preserve  the  same  value  but 
to  change  in  value;  such  numbers  (and  the  quantities  or 
magnitudes  which  they  measure)  are  called  variables. 

If  r  stands  for  the  distance  of  the  center  of  mass  of  the  earth  from 
the  center  of  mass  of  the  sun,  r  is  a  variable.  In  the  equation  s  = 
igt'  (the  law  of  falling  bodies),  if  i  be  the  elapsed  time,  s  the  distance 
traversed  from  rest  by  the  falling  body,  and  g  the  acceleration  due  to 
gravity,  then  s  and  t  are  variables  and  g  is  the  constant  32.2  feet  per 
second  per  second. 

The  following  are  constants:  Ratio  of  the  diameter  to  the  circum- 
ference in  any  circle;  the  electrical  resistance  of  pure  copper  at  60°  F. ; 
the  combining  weight  of  oxygen;  the  density  of  pure  iron;  the  velocity 
of  light  in  empty  space. 

The  following  are  variables:  the  pressure  of  steam  in  the  cyhnder 
of  an  engine;  the  price  of  wheat;  the  electromotive  force  in  an  alter- 
nating current;  the  elevation  of  groundwater  at  a  given  place;  the 
discharge  of  a  river  at  a  given  station.    When  any  of  these  magnitudes 


16 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


[§9 


are  assumed  to  be  measured,  the  numbers  resulting  are  also  variables. 
The  volume  of  the  mercury  in  a  common  thermometer  is  a  variable; 
the  mass  of  mercury  in  the  thermometer  is  a  constant. 

9.*  Graphical  Computation.  The  ordinary  operations  of  arith- 
metic, such  as  multiplication,  division,  involution  and  evolution, 
can  be  performed  graphically  as  explained  below.  The  graphical 
construction  of  products  and  quotients  is  useful  in  many  problems 
of  science.  The  fundamental  theorem  in  all  graphical  computa- 
tion is :  The  homologous  sides  of  similar  triangles  are  in  proportion. 
Its  application  is  very  simple,  as  wiU  appear  from  the  following 
work. 

Fboblem  1 :    To  compute  graphically  the  product  of  two  numbers. 
Let  the  two  numbers  whose  product  is  required  be  a  and  b.    On 
any  line  lay  off  the  unit        y  jj 
of  measurement,  01,  Fig.     i" 
11.    On    the    same    line, 
and,  of  course,  to  the  same 


, 

,f 

/ 

/ 
/ 

/ 

1 
1 

U) 

Ai 

r=c 

A- 

B 

/ 

/ 

(B) 

\l 

= 

AC 
OA 

/ 

/ 

B 

/ 

\^ 

O        1 

Fig.  11.  —  Graphical 
multiplication  by  proper- 
ties of  similar  triangles. 


5     6      7     8      9    10 


Fig.  12.— Method  of  graphical  mul- 
tiplication and  division  carried  out  on 
squared  paper.  The  figure  shows  1 . 9 
X4.4  =  8.4. 


scale,  lay  off  OA  equal  to  one  of  the  factors  a.  On  any  other 
line  passing  through  1  lay  off  a  Une  IB  equal  to  the  other  factor 
6.  Join  OB  and  produce  it  to  meet  AC  drawn  parallel  to  IB. 
Then  AC  is  the  required  product.    For,  from  similar  triangles. 


or 


AC:\B  =  0A:  01, 
AC  =  OA  X  IB. 


(1) 

(A) 


§9]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES         17 

AC  is  to  be  measured  with  the  same  scale  used  in  laying  off 
01,  OA,  and  IB.  The  number  of  unit's  in  AC  is  then  the  product 
of  a  by  6. 

It  is  obvious  that  the  angle  OiB  may  be  of  any  magnitude. 
Hence  it  may  conveniently  be  taken  a  right  angle,  in  which  case  the 
work  may  readily  be  carried  out  on  ordinary  squared  paper. 
Many  prefer,  however,  to  do  the  work  on  plain  paper,  la3dng  off  the 
required  distances  by  means  of  a  boxwood  triangular  scale.  If 
squared  paper  is  preferred  draw  the  two  lines  OX  and  OF  at 
right  angles  and  the  unit  line  If/,  as  shown  in  Fig.  12.'! 

In  the  exercises  that  follow  the  dimensions  are  given  in  inches. 
If  the  centimeter  scale  or  squared  paper  Form  M\  be  used,  use 
2  cer\timeters  everywhere  in  place  of  1  inch. 

Exercises 

1.  Find  graphically  the  product  of  1.63  by  2.78. 

Hird:  Choose  2  inches  to  represent  one  unit.  Draw  a  horizontal 
line  OA  5.56  inches  long.  Lay  off  the  distance  01  2  inches  in 
length.  Draw  IB  perpendicular  or  nearly  perpendicular  to  OA  and 
lay  off  IB  equal  to  3.26  inches  in  length.  Draw  OB.  Draw  AC 
parallel  to  IB.  Measure  AC.  One-half  of  the  length  of  AC  in  inches 
win  be  the  desired  product.  It  will  be  noticed  that  the  smaller  factor 
is  laid  off  on  IB. 

2.  Find  graphically  the  product  of  3.15  by  6.27.  Let  1  inch 
represent  one  unit. 

3.  Fmd  graphically  the  product  of  36.7  by  5.82. 

Hivi:  Find  the  product  of  3.67  by  5.82  and  then  move  the  decimal 
point  one  place  to  the  right. 

4.  Find  graphically  the  product  of  936  by  3.17. 

HiTii:  VmA  the  product  of  0.936  by  3.17  and  move  the  decimal 
point  three  places  to  the  right  in  the  result  obtained.  Let  2  inches 
represent  one  unit. 

5.  Fiud  graphically  the  product  of  9.36  by  7.23. 

Hint:  Ymd  the  product  of  0.936  by  0.723  and  move  the  decimal 
point  three  places  to  the  right  in  the  result  obtained.  Let  5  inches 
represent  one  unit. 

Problem  2 :  To  compute  graphically  the  quotient  of  two  numbers 
a  and  b.    Formula  (A)  above  can  be  written 


18  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§9 

From  this  it  is  seen  that  the  quotient  of  two  numbers  a  and  6  can 
readily  be  computed  graphically  by  use  of  Figs.  11  or  12. 

Exercises 

1.  Compute  graphically  the  quotient  of  1.33  divided  by  1.72. 
Hint:    Let  5  inches  represent  one  unit.     Lay  off  OA  equal  to  8.6 

inches.  Draw  AC  perpendicular  or  nearly  perpendicular  to  OA. 
Lay  off  AC  equal  to  6.65  inches.  Draw  OC.  Draw  IB  parallel  to 
AC.  One-fifth  the  number  of  inches  in  the  length  of  IB  is  the  required 
quotient. 

2.  Compute  graphically  the  quotient  of  7.32  divided  by  1.26. 
Hint:    Find  the  quotient  of  0.732  by  1.26,  using  5  inches  to  represent 

one  unit. 

3.  Compute  graphically  137  divided  by  732. 

Hint:  Calculate  1.37  divided  by  0.732  and  move  the  decimal  point 
one  place  to  the  left  in  the  result  obtained.  Use  5  inches  to  represent 
one  unit. 

Pboblem  3 :  To  compute  graphically  the  square  of  any  number  N. 
This  is  a  special  case  of  Problem  1,  when  the  two  factors  are 
equal. 

Exercises 

1.  Find  graphically  the  square  of  (o)  5;  (6)  3;  (c)  2, 

Hint:  In  finding  the  square  of  5,  first  find  square  of^O.S.  Let  10 
inches  represent  one  unit. 

2.  Find  graphically  the  square  of  93.6. 
Hint:    Find  the  square  of  0.936. 

3.  Find  graphically  the  square  of  0.0672. 
Hint:    Find  the  square  of  0.672. 

4.  Find  graphically  the  square  of  112. 
Hint:    Find  the  square  of  1.12. 

Phoblbm  4 :  To  compute  graphically  the  reciprocal  of  any  numherN. 
This  is  a  special  case  of  Problem  2,  when  the  dividend  is  1  and 
the  divisor  is  N. 

Exercises 

Find  graphically  the  reciprocals  of  the  following:  (o)  2;  (b)  3.5; 
(c)  12.3;  (d)  0.817.  , 

Peoblem    5:    To  compute  graphically  the  square  root  of  any 


VARIABLES  AND  FUNCTIONS  OF  VARIABLES 


19 


number  N.  On  OX,  Fig.  13,  lay  off  01  =  1  and  lA  =  N.  Upon 
OA  as  diameter  describe  a  semicircle  OCA.  At  1  erect  a  per- 
pendicular, IC,  to  OA.     Then  IC  is  the  square  root  of  lA. 

Another  construction  is  to  place  a  celluloid  triangle  in  the 
position  shown  in  Fig.  13,  so  that  the  two  edges  pass  through 
0  and  A  and  the  vertex  of  the  right  angle  Ues  on  the  line  1 U. 
Fig.  13  shows  the  construction  for  \/7. 


in' 

— 

u 

q 

g 

■^ 

-^ 

^ 

s 

//' 

■^- 

\ 

B 

^~ 

A 

X 

"< 

,  i 

/ 

. 

>    ( 

"^ 

w 

b 

/ 

r 

"■- 

, 

'- 

-._ 

/ 

1 

->. 

" 

~- 

/ 

f 

/ 

— 

— 

— 

Fig.  13. — Graphical  method  of  the  extraction  of  square  roots.    The 
figure  shows  Vt  =  2 .  Q5. 

Exercises 

Find  graphically  the  square  roots  of  the  following:  (o)  2;  (b)  3; 
(c)  5;  (d)  10;  (e)  932. 

Hivi:  In  part  (e)  find  the  square  root  of  9.32  and  move  the  decimal 
point  one  place  to  the  right  in  the  result  obtained. 

Problem  6:  To  compute  graphically  the  integral  powers  of 
any  number  N.  This  problem  is  solved  by  the  successive  applica- 
tion of  Problem  1  to  construct  N'^,  N^,  N*,  etc.,  and  of  Problem  2 
to  construct  iV"',  N~^,  N'^,  etc.  This  construction  is  shown  for 
the  powers  of  1.5  in  Fig.  14. 


Exercises 

1.  Compute  graphicaUy  (a)  (1.2)^;  (6)  (0.85)»;  (c)  (1.72)-2. 
Hint:    Let  5  inches  represent  one  unit. 


20 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


2.  Show  that  (1.05)^^  =  2.08,  so  that  money  at  5  percent  compoimd 
interest  more  than  doubles  itself  in  fifteen  years. 

Note:  The  work  is  less  if  (1.05)*  is  firstfound  and  then  this  result 
cubed. 

3.  From  the  following  outline  the  student  is  to  produce  a  complete 
method,  including  proof,  of  constructing  successive  powers  of  any 
number. 


R 

4 

/ 

/ 

/ 

/ 

/ 

/ 

// 

7 

3 

1 

/ 

/ 

A 

2 

/ 

// 

/ 

1 

// 

// 

/ 

1 

4 

.1 

1 

^ 

-3 
-4 

0  1       N     2  3  i 

Fig.  14. — Graphical  computation  of  (1.5)"  for  n  =  —4, 
0,  1,  2,  3,  4,  6. 


-3,  -2,  -1, 


Let  OA  (Fig.  15)  be  a  radius  of  a  circle  whose  center  is  0.  Let 
OB  be  any  other  radius  making  an  acute  angle  with  OA.  From  B 
drop  a  perpendicular  upon  OA,  meeting  the  latter  at  Ai.  From  Ai 
drop  a  perpendicular  upon  OB  meeting  OB  at  Ai.  From  .Aj  drop  a 
perpendicular  upon  OA  meeting  OA  at  A3,  and  so  on  indefinitely. 
Then,  if  OA  be  unity,  OAi  is  less  than  unity,  and  OAi,  OAs,  OAt 
.   .    .  are,  respectively,  the  square,  cube,  fourth  power,  etc..  of  OAi. 


§10]      VARIABLES  AND  FUNCTIONS  Of  VARIABLES        21 

Instead  of  the  above  construction,  erect  a  perpendicular  to  OB 
meeting  OA  produced  at  ai.  At  Oi  erect  a  perpendicular  meeting  OB 
produced  at  02,  and  so  on  indefinitely.  Then  if  OA  be  unity,  ai  is 
greater  than  unity  and  az,  03,  04,  .  are,  respectively,  the  square. 


Fig.  15.= — Graphical  computation  of  powers  of  a  number. 


cube,  etc.,  of  oi.  As  an  exercise,  construct  powers  of  4/5  and  of  2.5. 
4.  Show  that  the  successive  "treads  and  risers"  of  the  steps  of  the 
"stairways"  of  Fig.  16a  and  166  are  proportional  to  the  powers  of  r. 
The  figures  are  from  Milaukovitch,  Zeitschrift  fiir  Math,  und  Nat. 
Unterricht,  Vol.  40,  p.  329. 


Fig.  16. — Computation  of  ar,  or',  ar',   .    .    .  for  r  <  1  and  for  r  >  1. 


10.*  Double  Scales  for  Several  Simple  Algebraic  Functions.    We 

may  make  use  of  the  graphical  method  of  computation  explained 
above  to  construct  graphically  double  scales  representing  simple 


22 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§10 


algebraic  relations.  For  example,  we  may  construct  a  double 
scale  for  determining  the  square  of  any  desired  number.  Call 
OA  (see  Fig.  17)  the  scale  on  which  we  desire  to  read  the  number; 
call  OB  the  scale  on  which  we  read  the  square.  Let  us  agree  to 
lay  off  OA  as  a  uniform  scale,  using  01  as  the  unit  of  measure. 
Since  we  desire  to  read  opposite  0,  1,  2,  3,  .  .  .  of  the  uniform 
scale,  the  squares  of  these  numbers,  the  lengths  along  the  scale 
OB  must  be  laid  off  proportional  to  the  sqvare  roots  of  the  numbers 


Fig. 


-1 
17. 


01234567 

-Method  of  constructing  a  double  scale  of  squares  or  of 
square  roots. 


0,  1,  2,  3,  .  .  .  that  is,  the  square  root  of  any  length,  when 
laid  off  on  OB,  and  marked  with  the  symbol  of  the  original  length, 
will  he  opposite  the  square  root  of  that  number  on  OA. 

No  difficulty  need  be  experienced  in  carrying  out  the  actual  con- 
struction of  double  scales  representing  algebraic  relations,  either  by 
use  of  a  table  of  numerical  values  of  the  function  or  by  means  of 
graphical  construction.  As  a  less  laborious  method  of  graphically 
expressing  functional  relations  will  be  explained  in  the  next  chapter, 
the  matter  of  double  scales  will  not  be  discussed  further  at  this  place . 


CHAPTER  II 

RECTANGULAR  COORDINATES  AND  THE  STRAIGHT 

LINE 

11.  Statistical  Graphs.  Prom  work  in  elementary  algebra  the 
student  is  familiar  with  the  construction  of  statistical  graphs 
simUar  to  Figs.  18  and  19.  The  student  should  carefully  study 
the  construction  of  these  two  graphs.  In  Fig.  18,  the  point  at 
the  center  of  any  small  circle  represents  the  maximum  temperature 
(or  the  minimum  temperature)  on  a  particular  day.  This  circle 
is  joined  by  a  straight  hne  to  the  circle  representing  the  maximum 
(or  minimum)  temperature  on  the  next  day,  and  so  on.  The 
lines  joining  the  circles  enable  the  eye  to  foUow  at  a  glance  the 
changes  of  temperature  for  the  entire  month.  However,  a  point 
on  a  line  between  two  circles  has  no  meaning,  because  a  point 
on  the  horizontal  scale  between  two  consecutive  points  has  no 
meaning,  for  of  course  there  is  but  one  daily  maximum  for  each 
day.  The  student  should  especially  note  that  the  ratio  of  the 
distance  on  the  horizontal  scale  representing  days  to  the  distance 
on  the  vertical  scale  representing  a  chajige  of  one  degree  in 
temperature  is  so  chosen  as  to  make  the  fluctuations  in  the 
temperatures  stand  out  prominently.  In  constructing  statistical 
graphs,  the  student  should  always  choose  this  ratio  so  that  the 
graph  will  clearly  convey  its  intended  meaning. 

Smooth  curves  are  drawn  through  the  plotted  points  of  Fig.  19 
(not  straight  lines  as  in  Fig.  18)  because  in  this  case  intermediate 
points  have  meaning;  they  represent  temperatures  at  various 
times  of  the  day. 

Fig.  20  is  a  barograph,  or  autographic  record  of  the  atmospheric 

pressure  recorded  November  24,  1907,  during  a  balloon  journey 

from  Frankfort  to  Marienburg  in  West  Prussia.    The  zero  of  the 

scale  of  pressure  does  not  appear  in  the  diagram.    Note  also 

-  that  the  scale  of  pressure  is  an  inverted  scale,  increasing  downward. 

23 


24         ELEMENTARY  MATHEMATICAL  ANALYSIS        t§ll 


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^isqusiqcj  saajSaQ  — siTHEjaamaj; 


§11] 


RECTANGULAR  COORDINATES 


25 


The  scale  of  time  is  an  algebraic  scale,,  the  zero  of  which  may  be 
arbitrarily  selected  at  any  convenient  point.  The  scale  of  pres- 
sure is  an  arithmetical  scale.    The  zero  of  the  barometric  scale 


'i70 

— 

— 

— 

— 

— 

— ~ 

— 

— 

— 

— 

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~ 

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-_ 

r:^ 

rt: 

— 

~~ 

~ 

— 

~ 

~ 

— 

~ 

§ 

Houriy_Air  Temperatures 
at 
Madison  Wisconsin 
May,14.i910 

and 
Oct..  10. 1910 

0)  . 

|io 

0 

2          4 

3 

8          10        12_       2          4        _ 

i 

8         10        1? 

Til 

ne 

oil 

iaV 

Hours 

1/ 

Fig.  19. — Hourly  air  temperatures. 


corresponds  to  a  perfect  vacuum — no  less  pressure  and  hence, 
in  this  case,  no  negative  value  exists. 

Fig.  21  is  a  graphical  time-table  of  certain  passenger  trains  be- 
tween Chicago  and  Minneapolis.    The  curves  are  not  continuous. 


s 

g 

-3 

^ 

s 

=1 

Fig.  20. — Barograph  taken  during  a  balloon  journey.     The  vertical 
scale  is  atmospheric  pressure  in  miUimeters  of  mercury. 

as  in  the  case  of  the  barograph,  but  contain  certain  sudden  jumps. 
What  is  the  meaning  of  these?  What  indicates  the  speed  of  the 
trains?  Where  is  the  fastest  track  on  this  railroad?  What 
shows  the  meeting  jxrint  of  trains? 


26 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§11 


If  the  diagram,  Fig.  21,  he  wrapped  around  a  vertical  cylinder  of 
such  size  that  the  two  midnight  lines  coincide,  then  each  train 
line  may  be  traced  through  continuously  from  terminus  to  terminus. 
Functions  having  this  remarkable  property  are  said  to  be  peri- 
odic. In  the  present  case  the  trains  run  at  the  same  time  every 
day,  that  is,  periodically.  In  mathematical  language,  the  po- 
sition of  the  trains  is  said  to  be  a  periodic  function  of  the  time. 


Chicago 


£au  Olaire 

Menomoaie 

Hudson 
St  Paal 
MinneapolieJf 


10 


12      2 
A.M.  Noon  P.M. 

Fig.  21. — Graphical  time  table  of  certain  passenger,  trains  between 
Chicago  and  Minneapolis. 

Fig.  22  represents  the  fluctuation  of  the  elevation  of  the  ground- 
water at  a  certain  point  near  the  sea  coast  on  Long  Island.  The 
fluctuations  are  primarily  due  to  the  tidal  wave  in  the  near-by 
ocean.  The  curve  is  continuous.  Is  the  curve  periodic?  What 
indicates  the  rate  of  change  in  the  elevation  of  the  ground- water? 
When  is  the  elevation  changing  most  rapidly?  When  is  it 
changing  most  slowly? 

Fig.  23  represents  the  functional  relation  between  the  amount  of 
a  domestic  money  order  and  the  fee.    This  is  an  excellent  illustra- 


§12] 


RECTANGULAR  COORDINATES 


27 


tion  of  a  discontinuous  function.  On  account  of  the  sudden 
jumps  in  the  values  of  the  fee,  the  fee,  as  explained  in  the  preceding 
chapter,  is  said  to  be  a  discontinuous  function  of  the  amount  of 
the  order. 


Fig.  22. — Upper  curve,  elevation  of  water  in  a  well  on  Long  Island. 
Lower  curve,  elevation  of  water  in  the  nearby  ocean. 

12.  Suggestions  on  the  Construction  of  Graphs.  Two  kinds  of 
rectangular  coordinate  paper  have  been  prepared  for  use  with  this 
book.  Form  Ml  is  ruled  in  centimeters  and  fifths.  Form  M2 
is  ruled  without  major  divisions  in  uniform  1/5-inch  intervals. 

It  is  a  mistake  to  assume  that  more  accurate  work  can  be  done  on 
finely  ruled  than  on  more  coarsely  ruled  squared  paper.     Quite  the 


28 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§12 


contrary  is  the  case.  Paper  ruled  to  1/20-inoh  intervals  does  not  per- 
mit interpolation  within  the  small  intervals  while  paper  ruled  to  1/10 
or  1/5-inch  intervals  permits  accurate  interpolation  to  one-tenth  of  the 
smallest  interval.  Form  Ml  is  ruled  to  2-mm.  intervals,  and  is  fine 
enough  for  any  work.  The  centimeter  unit  has  the  very  considerable 
advantage  of  permitting  twenty  of  the  units  within  the  width  of  an 
ordinary;  sheet  of  letter  paper  (SJ  X  11  inches)  while  seven  is  the 
largest  number  of  inch  units  available  on  such  paper. 

In  order  to  secure  satisfactory  results,  the  student  must  recog- 
nize that  there  are  several  varieties  of  statistical  graphs,  and  that 
each  sort  requires  appropriate  treatment. 


-50 


MO  1 


-30 


r-20 


HlO 


1. 1. .Ill 


j_ 


_L 


J. 


_!_ 


I 


_L 


J_ 


10 


90 


20         80         40         50         60         70  *      80 
Amount  of  the  Money  Order  in  Dollars 
Fig.  23. — The  graph  of  a  discontinuous  function. 


1.  It  is  possible  to  make  a  useful  graph  when  only  one  variable 
is  given.  Thus  Table  I  gives  the  ultimate  tensile  strength  of 
various  materials. 

A  graph  showing  these  results  is  given  in  Fig.  24.  There  are 
two  practical  ways  of  showing  the  numerical  values  pertaining 
to  each  material,  both  of  which  are  indicated  in  the  diagram; 
either  rectangles  of  appropriate  height  may  be  erected  opposite  the 
name  of  each  material,  or  points  marked  by  circles,  dots  or  crosses 
may  be  located  at  the  appropriate  height.  It  is  obvious  in  this 
case  that  a  smooth  curve  should  not  be  drawn  through  these  points 
— such  a  curve  would  be  quite  meaningless.    In  this  case  there 


§12]  RECTANGULAR  COORDINATES  29 

Table  I. — Ultimate  Tensile  Strength  of  Various  Materials 


Material 

Tensile  strength, 
tons  per  square  inch 

Hard  steel 

50.0 
30.0 
25.0 
21.5 
16.0 
12.0 
11.0 
10.0 
5.0 

Wrought  iron 

Drawn  brass 

Cast  brass 

Timber,  with  grain 

are  not  two  scales,  but  merely  the  single  vertical  scale.  The  hori- 
zontal axis  bears  merely  the  names  of  the  different  materials 
and  has  no  numerical  or  quantitative  signifioance.  The  result 
is  obviously  not  the  graph  of  a  function,  for  there  are  not  two 
variables,  but  only  one.  The  graph  is  merely  a  convenient  ex- 
pression for  certain  discrete  and  independent  results  arranged 
in  order  of  descending  magnitude. 

2.  It  is  possible  to  have  a  graph  involving  two  variables  in 
which  it  is  either  impossible  or  undesirable  to  represent  the  graph 
by  a  continuous  curve  or  line.  For  example.  Fig.  18  is  a  graph 
representing  the  maximum  temperature  on  each  day  of  a  certain 
month.  Because  there  is  only  one  maximum  temperature  on 
each  day,  the  value  corresponding  to  this  should  be  shown  by  an 
appropriate  rectangle,  or  by  marking  a  point  by  a  circle,  or  by  a 
dot  or  cross,  as  in  the  preceding  case.  A  continuous  curve 
through  these  points  has  no  meaning.  The  horizontal  scale  may 
be  marked  by  the  names  of  the  days  of  the  week  or  by  numbers, 
but  in  either  case  the  horizontal  line  is  a  true  scale,  as  it  cor- 
responds to  the  lapse  of  the  variable  time.  Sometimes,  as  in  Fig. 
18,  graphs  of  this  kind  are  represented  by  marking  the  appropriate 
points  by  dots  or  circles  and  then  connecting  the  successive  points 
by  straight  lines.  These  lines  have  no  special  meaning  in  such 
a  case,  but  they  aid  the  eye  in  following  the  succession  of  separate 
points. 


30 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§12 


If  a  graph  be  made  of  the  noonday  temperatures  of  each  day 
of  the  same  month  referred  to  in  Fig.  18,  one  of  the  same  methods 
indicated  above  would  be  used  to  represent  the  results;  that  is, 
either  rectangles,  marked  points,  or  marked  points  joined  by  lines. 
Although  a  smooth  curve  drawn  through  the  known  points  would 
have  a  meaning  (if  correct),  it  is  obvious  that  the  noonday 
temperatures  alone  are  not  sufficient  for  determining  its  form. 
In  all  such  cases  a  smooth  curve  should  not  be  drawn. 

3.  If  the  data  are  reasonably  sufficient,  a  smooth  curve  may, 
and  often  should,  be  drawn  through  the  known  points.    Thus  if 

the  temperature  be  observed 
every  hour  of  the  day  and  the  re- 
sults be  plotted,  a  smooth  curve 
drawn  carefully  through  the 
plotted  points  will  probably  very 
accurately  represent  the  un- 
known temperatures  at  interme- 
diate times.  The  same  may 
safely  be  done  in  exercises  (3) 
and  (4)  below.  In  scientific 
work  it  is  desirable  to  mark  by 
circles  or  dots  the  values  that 
are  actually  given  to  distinguish 
them  from  the  intermediate 
values  "guessed"  and  repre- 
sented by  the  smooth  curve. 

In  addition  to  the  above 
suggestions,  the  student  should 
adhere  to  the  following  instruc- 
tions : 

4.  Every    graph    should     be 

marked  with  suitable  numerals 

along  both  numerical  scales. 

5.  Each  scale  of  a  statistical  graph  should  bear  in  words  a 

description  of  the  magnitude  represented  and  the  name  of  the 

unit  of  measure  used.    These  words  should  be  printed  in  drafting 

letters  and  not  written  in  script. 


•9 

2 

H 

1 

JS!a(\ 

Id- 

1 

', 

\ 

i2<ift 

\ 

H 

-! 

t-1 

M 

a 

t- 

OS 

^ 

a  in 

■ 

" 

1- 

r" 

- 

~ 

n 

. 

_i 

n    D 


Fig.  24. — Graph  showing 
tensile  strength  of  certain  struc- 
tural materials. 


§12] 


RECTANGULAR  COORDINATES 


31 


6.  Each  graph  should  bear  a  suitable  title  telling  exactly  what  is 
represented  by  the  diagram. 

7.  The  selection  of  the  units  for  the  horizontal  and  vertical 
scales  is  an  important  practical  matter  in  which  common  sense 
must  control.  It  is  obvious  that  in  the  third  exercise  given 
below  1  cm.  =1  foot  draft  for  the  horizontal  scale,  and  1  cm. 
=  100  tons  for  the  vertical  scale  will  be  units  suitable  for  use  on 
form  Ml. 

Further  instruction  in  practical  graphing  is  given  in   §33. 

Exercises 

1.  Draw  a  statistical  graph  from  the  data  given  in  the  following 
table.  See  Mg.  18.  Represent  the  plotted  points  by  small  distinct 
points,  not  by  circles. 


Maximum  and  Minimum  Temperatures  at  Madison,  Wisconsin, 
FOR  October,  1910 


Date 

Max.  temp., 

Mm.  temp., 
°F. 

Date 

Max.  temp., 
"F. 

Min.  temp., 
"F. 

1 

68 

55 

17 

81 

53 

2 

71 

^8 

18 

81 

57 

3 

75 

58 

19 

69 

45 

4 

68 

55 

20 

45 

40 

5 

62 

53 

21 

49 

41 

6 

58 

45 

22 

52 

34 

7 

66 

43 

23 

60 

37 

8 

68 

47 

24 

60 

49 

9 

60 

44 

25 

52 

44 

10 

67 

42 

26 

60 

40 

11 

75 

49 

27 

42 

32 

12 

61 

46 

28 

35 

30 

13 

69 

45 

29 

38 

26 

14 

73 

52 

30 

60 

31 

15 

76 

50 

31 

63 

39 

16 

80 

56 

2.  Draw  a  statistical  graph  for  the  data  given  in  the  following  table. 
See  Fig.  19. 


32 


ELEMENTARY  MATHEMATICAL  ANALYSIS       [§12 


Hourly  Air  Temperatures  at  Madison,  Wisconsin,  Mat  14, 
1910;  October  10,  1910 


Time 

May  14, 

1910, 

temp.,  "  F. 

Oct.  10, 

1910, 

temp.,  "  F. 

Time 

May  14, 

1910, 

temp.,  °  F. 

Oct.  10, 

1910, 

temp.,  °  F. 

1  a.  m. 

37 

44 

1  p.  m. 

58 

65 

2  a.  m. 

35 

44 

2  p.  m. 

61 

66 

3  a.  m. 

35 

44 

3  p.  m. 

63 

67 

4  a.  m. 

34 

43 

4  p.  m. 

62 

67 

5  a.  m. 

34 

43 

5  p.  m. 

62 

65 

6  a.  m. 

35 

42 

6  p.  m. 

61 

62 

7  a.  m. 

37 

43 

7  p.  m. 

69 

57 

8  a.  m. 

42 

45 

8  p.  m. 

66 

65 

9  a.  m. 

46 

51 

9  p.  m. 

53 

55 

10  a.  m. 

49 

65 

10  p.  m. 

50 

66 

11  a.  m. 

54 

60 

11  p.  m. 

47 

62 

12  a.  m. 

56 

62 

12  p.  m. 

45 

51 

3.  At  the  following  drafts  a  ship  has  the  displacements  stated : 

Draft  in  feet,  h , 

15 

12 

9 

6.3 

Displacement  in  tons,  T 

2096 

1512 

1018 

586 

Plot  on  squared  paper.     What  are  the  displacements  when  the 
drafts  are  11  and  13  feet,  respectively? 

4.  The  following  tests  were  made  upon. a  steam  turbine  generator: 


Output  in  kilowatts,  K.. . 

1,190 

995 

745 

498 

247 

Weight,  pounds  of  steam 
consumed  per  hour,  W. 

23,120 

20,040 

16,630 

12,560 

8,320 

Plot  on  squared  paper.  What  are  the  probable  values  of  K  when 
W  is  22,000  and  also  when  W  is  11,000? 

6.  The  average  temperature  at  Madison  from  records  taken  at  7 
a.  m.  daily  for  30  years  is  as. follows: 

Jan.  1,  14.0.°  F. 
Feb.  1,15.1. 
Mar.  1,  35.2. 
Apr.  1,  40.0. 
May  1,  53.9. 
June  1,  63.^. 

Make  a  suitable  graph  of  these  results  on  squared  paper. 


July 

1,  67.5. 

Aug. 

1,  64.0. 

Sept. 

1,  55.4. 

Oct. 

1,  44.1. 

Nov. 

1,  30.0. 

Dec. 

1,  18.3. 

§13] 


RECTANGULAR  COORDINATES 


33 


13.  Rectangular  Cobrdinates.  Two  intersecting  algebraic 
scales,  with  their  zero  points  in  common,  may  be  used  as  a  system 
of  latitude  and  longitude  to  locate  any  point  in  their  plane.  The 
student  should  be  familiar  with  the  rudiments  of  this  method 
from  the  graphical  work  of  elementary  algebra.  The  scheme  is 
illustrated  in  its  simplest  form  in  Fig.  25,  where  one  of  the  hori- 
zontal lines  of  a  sheet  of  squared  paper  has  been  selected  as  one 
of  the  algebraic  scales  and  one  of  the  vertical  lines  of  the  squared 
paper  has  been  selected  for  the  second  algebraic  scale.  To  locate 
a  given  point  in  the  plane  it  is  merely  necessary  to  give,  in  a 


Y 

4 

'■7 

1 

Pr 

( 

iVzZH 

) 

Pi 

( 

-3, 

2) 

?, 

II 

I 

X' 

D 

X 

-: 

-2 

- 

0 

1 

2 

3 

i 

5 

.^ 

P> 

(- 

2.- 

1) 

-? 

I] 

I 

IV 

-3 

Y 

Pi 

(2,-S 

) 

^ 

Fig.  25. — Rectangular  coordinates. 


suitable  unit  of  measiu'e  (as  centimeter,  inch,  etc.),  the  distance 
of -the  point  to  the  right  or  left  of  the  vertical  scale  and  its  distance 
above  or  below  the  horizontal  scale.  Thus  the  point  Pi,  in  Fig. 
25,  is  2j  units  to  the  right  and  3j  units  above  the  standard 
scales.  P2  is  3  units  to  the  left  and  2  units  above  the  standard 
scales,  etc.  Of  course  these  directions  are  to  be  given  in  mathe- 
matics by  the  use  of  the  signs  "-)-"  and  "  — "  of  the  algebraic 
scales,  and  not  by  the  use  of  the  words  "right"  or  "left,"  "up" 
or  "down."  The  above  scheme  corresponds  to  the  location  of  a 
place  on  the  earth's  surface  by  giving  its  angular  distance  in 
3 


34  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§13 

degrees  of  longitude  east  or  west  of  the  standard  meridian,  and 
also  by  giving  its  angular  distance  in  degrees  of  latitude  north 
or  south  of  the  equator. 

The  sort  of  latitude  and  longitude  that  is  set  up  in  the  manner 
described  above  is  known  in  mathematics  as  a  system  of  rectangu- 
lar coordinates.  It  has  become  customary  to  letter  one  of  the 
scales  XX',  called  the  X-axis,  and  to  letter  the  other  YY',  called 
the  Y-axis.  In  the  standard  case  these  are  drawn  to  the  right 
and  left,  and  up  and  down,  respectively,  as  shown  in  Fig.  25. 
The  distance  of  any  point  from  the  F-axis,  measured  parallel  to 
the  X-axis,  is  called  the  abscissa  of  the  point.  The  distance  of 
any  point  from  the  X-axis,  measured  parallel  to  the  F-axis,  is 
called  the  ordinate  of  the  point.  Collectively,  the  abscissa  and 
ordinate  are  spoken  of  as  the  coordinates  of  the  point.  Abscissa 
corresponds  to  the  longitude  and  ordinate  corresponds  to  the 
latitude  of  the  point,  referred  to  the  X-axis  as  equator,  and  to 
the  F-axis  as  standard  meridian.  In  the  standard  case,  abscissas 
measured  to  the  right  of  YY'  are  reckoned  positive,  those  to  the 
left,  negative.  Ordinates  measured  up  are  reckoned  positive, 
those  measured  down,  negative. 

Rectangular  coordinates  are  frequently  called  Cartesian  co- 
ordinates, because  they  were  first  introduced  into  mathematics 
by  Ren6  Descartes  (1596-1650). 

The  point  of  intersection  of  the  axes  is  lettered  0  and  is  called 
the  origin.  The  four  quadrants,  XOY,  YOX',  X'OY',  Y'OX, 
are  called  the  first,  second,  third,  and  fourth  quadrants,  respectively. 

A  point  is  designated  by  writing  its  abscissa  and  ordinate  in  a 
parenthesis  and  in  this  order:  Thus,  (3,  4)  means  the  point 
whose  abscissa  is  3  and  whose  ordinate  is  4.  Likewise  (—3,  4) 
means  the  point  whose  abscissa  is  (—3)  and  whose  ordinate  is 
(+4). 

Abscissas  are  usually  represented  by  the  letter  x  and  ordinates 
are  usually  represented  by  the  letter  y.  Thus  the  point  whose 
abscissa  is  3  and  whose  ordinate  is  4,  may  be  described  as  the 
point  (3,  4),  or  equally  well  as  the  point  x  =  3,  y  —  4. 

Unless  the  contrary  is  expUcitly  stated,  the  scales  of  the  eo- 
■  ordinate  axes  are  assumed  to  be  straight  and  uniform  and  to  inter- 
sect at  right  angles.    Exceptions  to  this  are  not  uncommon. 


§14]  RECTANGULAR  COORDINATES  35 

Exercises 

On  suitable  squared  paper,  select  and  mark  a  horizontal  line  as  the 
X-axis  (or  axis  of  abscissas)  and  select  and  mark  a  vertical  line  as  the 
F-axis  (or  axis  of  ordinates).  Select  and  mark  a  suitable  unit  of 
measure  on  each  axis,  for  example  1  centimeter  or  1/2  inch. 
Then  locate  the  points  whose  coordinates  are  given  in  the  following 
exercises. 

1.  Draw  the  coordinate  axes  on  squared  paper  and  locate  the  points 
(3,  3),  (2,  2),  (1,  1),  (0,  0),  (-1,  -1),  (-2,  -2),  (-3,  -3). 

2.  Draw  the  axes  and  locate  the  points  (2,  3),  (—2,  3),  (  —  2,  —3), 
(2,  -3). 

3.  Draw  the  coordinate  axes  and  locate  the  points  (5,  0),  (4,  3), 
(3,  4),  (0,  5),  (-3,  4),  (-4,  3),  (-5,  0),  (-4,  -3),  (-3,  -4),  (0,  -5), 
(3,  -4),  (4,  -3). 

4.  Draw  suitable  axes  and  locate  the  points  (  —  3,  —5),  (—2,  —3), 
(-1,  -1),  (0,  1)  (1,  3),  (2,  5),  (3,  7),  (4,  9). 

A  brief  way  of  describing  a  ^et  of  points  is  to  place  the  abscissas  and 
ordinates  in  tabular  form,  indicating  abscissas  by  the  letter  x  and 
indicating  ordinates  by  the  letter  y,  as  follows : 

a;   I  -3'  -2  -10  1  2  3  4 

y   \  -5-3-113579 

14.  Mathematical,  or  Non-statistical  Graphs. — Instead  of  the 
expressions  "abscissa  of  a  ■point,"  or  "ordinate  of  a  point,"  it  has  be- 
come usual  to  speak  merely  of  the  "x  of  a  point,"  or  of  the  "y  of 
a  point,"  since  these  distances  are  conventionally  represented  by 
the  letters  x  and  y,  respectively.  If  we  impose  certain  conditions 
upon  X  and  y,  then  it  will  be  found  that  we  have,  by  that  very  fact, 
restricted  the  possible  points  of  the  plane  located  by  them  to  a 
certain  array,  or  set  of  points,  and  that  all  other  points  of  the 
plane  fail  to  satisfy  the  conditions  or  restrictions  imposed. 

It  is  obvious  that  the  command,  "Find  the  place  whose  latitude 
equals  its  longitude,"  does  not  restrict  or  confine  a  person  to  a  par- 
ticular place  or  point.  The  places  satifying  this  condition  are 
unlimited  in  number.  We  indicate  all  such  points  by  drawing 
a  line  bisecting  the  angles  of  the  first  and  third  quadrants;  at  all 
points  on  this  line  latitude  equals  longitude.  We  speak  of  this 
line  as  the  locus  of  the  point  satisfying  the  conditions.  We  might 
describe  the  same  locus  by  saying  "the  y  of  each  point  of  the 


36 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§14 


locus  equals  the  x"  or,  with  the  maximum  brevity,  simply, write 
the  equation  "y  =;  x."  The  equation  "y  =  x"  is  called  the 
equation  of  the  locus,  and  the  line  is  called  the  locus  of  the 
equation. 

It  is  of  the  utmost  importance  to  be  able  readily  to  interpret  any 
condition  imposed  upon,  or,  what  is  the  same  thing,  any  relation 
between  variables,  when  these  are  given  in  words.  It  will  greatly 
aid  the  beginner  in  mastering  the  concept  of  what  is  meant  by  the 
term  fimction  if  he  will  try  to  think  of  the  meaning  in  words  of  the 
relations  commonly  given  by  equations,  and  vice  versa.  The 
very  elegance  and  brevity  of  the  mathematical  expression  of  rela- 
tions by  means  of  equations,  tends  to  make  work  with  them  formal 


y 

"h 

i 

2 

^ 

/ 

/ 

1 

/■ 

X' 

0 

/ 

X 

1  D 

/    ^' 

Fig.  26. — The  straight  line  y  =  x. 


Fig.  27.— The  straight  line 
y  =  2x. 


and  mechanical  unless  care  is  taken  by  the  beginner  to  express  in 
words  the  ideas  and  relations  so  briefly  expressed  by  the  equa- 
tions. Unless  expressed  in  words,  the  ideas  are  liable  not  to  be 
expressed  at  all. 

The  equation  of  a  curve  is  an  equation  satisfied  by  the  co- 
ordinates of  every  point  of  the  curve  and  by  the  coordinates  of  no 
other  point. 

The  graph  of  an  equation  is  the  locus  of  a  point  whose  coordi- 
nates satisfy  the  equation. 

Illustration  I.     Find  the  equation  of  the  Une  of  Fig.  26.     The  Une 


§14] 


RECTANGULAR  COORDINATES 


37 


of  this  figure  states  that  the  y  of  any  point  of  the  line  equals  the  x  of 
that  point.     Hence  the  equation  of  the  line  isy  =  x. 

Illustration  2.  The  line  of  Fig.  27  is  drawn  through  the  origin  and 
the  point  (1,  2).  Find  the  equation  of  the  line.  Let  OB  and  DP  be 
the  abscissa  and  ordinate,  respectively,  of  any  point  on  the  line.  Then 
from  similar  triangles  OPD  and  OPil,  DP:OD  =  2  : 1,  or  y  :  a;  =  2: 1, 
or  y  =  2x,  which  is  the  equation  of  the  line. 

Exercises 


What  is  its 


What  is 


What  is 


1.  Draw  a  hne  through  the  origin  and  the  point  (1,  3). 
equation? 

2.  Draw  a  line  through  the  origin  and  the  point  (1,  -f)- 
its  equation? 

3.  Draw  a  line  through  the  origin  and  the  point  (1,  —1). 
its  equation?     Draw  a  line  through  the 
origin  and  the  point  (1,  —2).     What  is  its 
equation? 

4.  Draw  loci  for  the  following  and  show 
that  each  locus  is  a  straight  line  passing 
through  the  origin:  (a)  The  ordinate  of 
any  point  of  a  certain  locus  is  twice  its 
abscissa;  (b)  the  x  of  every  point  of  a  cer- 
tain locus  is  half  its  y;  (c)  the  yoia.  point  is 
1/3  of  its  x;  (d)  a  point  moves  in  such  a 
way  that  its  latitude  is  always  treble  its 
longitude;  (e)  the  sum  of  the  latitude  and 
longitude  of  a  point  is  zero;  (/)  a  point 

moves  so  that  the  difference  in  its  latitude  and  longitude  is  always 
zero. 

Hint:  In  part  (a)  let  Pi  (Fig.  28)  be  any  point  on  the  locus  and 
let  Pi  be  any  second  point  on  the  locus. 

Draw  OPi  and  OPi]  draw  PiDi  and  P2D2  perpendicular  to  OX.  By 
the  conditions  of  the  problem  PiDi  =  20Di  and  P2D2  =  20Di. 
Hence 

PiDi  _P,D2 
ODi       OD2' 

and  the  triangles  OPiDi  and  OP2D2  are  similar.  Then  the  angles 
PiODi  and  P2OD2  are  equal.  Hence  OPi  and  OP2  coincide  in  direction 
and  0,  Pi,  and  P2,  are  upon  a  straight  line. 

5.  Draw  the  locus:  Beginning  at  the  point  (1,  2),  a  point  moves  so 


Fig.  28. — Diagram 
for  exercise  4  (o)  §14  and 
,  exercise  3  §15. 


38 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§15 


that  its  gain  ia  latitude  is  always  twice  as  great  as  its  gain  in  longitude. 
Show  that  the  locus  is  a  straight  line. 

6.  A  point  moves  so  that  its  latitude  is  always  greater  by  2  units 
than  three  times  its  longitude.  Write  the  equation  of  the  locus  and 
construct-     Show  that  the  locus  is  a  straight  line. 

15.  Slope.  The  slope  of  a  straight  line  is  defined  to  be  the 
change  in  y  for  an  increase  in  x  equal  to  1.  It  will  be  represented 
in  this  book  by  the  letter  to.     In  Fig.  26  the  line  OP  has  slope  1  and 

in  Fig.  27  the  line  OP 
has  slope  2.  Also  in 
Fig.  29  the  line  A  has 
the  slope  to  =  1.5,  for 
it  is  seen  that  at  any 
point  of  the  linb  the 
ordinate  y  gains  1.5 
units  for  an  increase 
of  1  in  X.  The  line  B, 
parallel  to  the  line  A, 
is  also  seen  to  have  the 
slope  equal  to  1.5. 
The  equation  of  the 
line  A  is  obviously  y 
=  1.5a;.  In  the  same 
figure  the  slope  of  the 
line  C  is  (  —  2),  for  at 
any  point  of  this  line 
the  ordinate  2/  decreases 
2  units  for  an  increase 
in  X  equal  to  1.  The  equation  of  the  line  C  is  obviously 
y  =  —2x.    Line  D,  parallel  to  line  C,  also  has  slope  (  —  2). 

If  h  be  the  change  in  y  for  an  increase  of  x  equal  to  k,  then  the 
slope  TO  is  the  ratio  h/k.  Hence  the  practical  method  of  determin- 
ing the  slope  of  a  line  drawn  upon  squared  paper  is:  Select  two 
convenient  points  on  the  line  rather  far  apart,  and  divide  the  change 
in  y  by  the  increase  in  x. 

The  technical  word  slope  differs  from  the  word  slope  or  slant  in 
common  language  only  in  the  fact  that  slope,  in  its  technical  use, 
is  always  expressed  as  the  ratio  of  two  algebraic  numbers.     In 


Y 

U 

C 

1 

s 

B 

A 

\ 

\ 

/ 

/ 

\ 

I 

\ 

m 

.- 

2 

4n 

=  1 

■y 

\ 

\ 

/ 

/ 

t 

m\ 

B- 

2\ 

(^ 

/ 

1 

/ 

\ 

\, 

/ 

\ 

\ 

-> 

/ 

^m 

=  + 

1.. 

\, 

\ 

/ 

/ 

1 

\ 

\ 

/ 

/ 

\, 

/ 

\; 

^m 

-1 

.5 

X 

/ 

^ 

/ 

X 

-5 

-4 

-3 

-2 

/ 

i^-l 

\ 

/ 

\ 

1 

2 

3 

4 

6 

/ 

\ 

-1 

\ 

m 

-- 

2 

/ 

/ 

V 

\ 

' 

/ 

\ 

\ 

/ 

t 

\ 

s 

/ 

/ 

-9 

\ 

\ 

/ 

/ 

V 

/ 

f 

-\ 

\ 

/ 

/ 

\ 

\ 

/ 

/ 

-5 

Y 

s 

..S 

Fig.  29.- 


-Lines  of  slope  (1 . 5)  and  of  slope 
(-2). 


§16]  RECTANGULAR  COORDINATES  39 

common  language  we  speak  of  a  "slope  of  1  in  10,"  or  a  "grade 
of  50  feet  per  mile,"  etc.  In  mathematics  the  equivalents  are 
"slope  =  1/10,"  "slope  =  50/5280,"  etc. 

As  already  indicated,  the  definition  of  slope  requires  us  to  speak 
in  mathematics  of  positive  slope  and  negative  slope.  A  line  of  posi- 
tive slope  extends  upward  with  respect  to  the  standard  direction 
OX  and  a  line  of  negative  slope  extends  downward  with  reference 
toOZ. 

In  a  similar  way  we  may  speak  of  the  slope  of  any  curve  at  a 
given  point  on  the  curve,  meaning  thereby  the  slope  of  the  tangent 
line  drawn  to  the  curve  at  that  point. 

Exercises 

1.  Give  the  slopes  of  the  lines  in  exercises  1  to  6  of  the  preceding  set 
of  exercises. 

2.  Draw  y  =  x;  y  =  2x;  y  =  Zx;  y  =  -^-t  y  =  ^,  y  =  ^,  y  =  —  2x; 

y  =  — 3z;  y  =  Ox. 

3.  Prove  that  y  =  mx  always  represents  a  straight  line,  no  matter 
what  value  m  may  have.     Hint:  Make  use  of  Fig.  28. 

16.  Equation  of  a  Straight  Line.  Intercepts. — In  Fig.  30,  the 
line  MN  expresses  that  the  ordinate  y  is,  for  all  points  on  the  line, 
always  3  times  the  abscissa  x,  or  it  says  that  y  =  3a;.  The  line 
HK  is  2  units  higher  than  MN,  so  that  it  states  that  "2/  is  2  more 
than  3a;."  Thus  the  line  HK  has  the  equation  y  =  Zx  +  2.  In 
Fig.  29  the  line  5  is  2  units  higher  than  the  line  y  =  1.5a;,  hence 
its  equation  is  y  =  1.5x  -{-  2.  The  line  D  is  2  units  lower  than 
the  line  C,  whose  equation  is  y  =  —2x,  hence  the  equation  of 
Disy  =  -2x  -  2. 

In  general,  since  y  =  mx  is  always  a  straight  line,^  then  y  = 
ma;  +  6  is  a  straight  line,  for  the  y  of  this  locus  is  merely,  in  each 
case,  the  y  of  the  former  increased  by  the  constant  amount  6  (which 
may,  of  course,  be  positive  or  negative).  Therefore,  y  —  mx  +  6 
is  a  line  parallel  to  y  =  mx.  The  line  y  =  mx  +  6  is  6  units 
higher  than,  or  above,  the  line  y  =  mx  ii  b  stands  for  a  positive 
number  and  the  line  y  =  mx  +  b  is  b  units  lower  than,  or  below 
the  line  y  =  mx  if  6  stands  for  a  negative  number.    The  distance 

1  See  exercise  3,  §15,  above. 


40 


ELEMENTARY  MATHEMATICAL  ANALYSIS        l§16 


OB  (Fig.  30)  is  equal  to  b,  and  is  called  the  Y-intercept  of  the 
graph.  The  distance  OA  is  equal  to  —  b/m,  for  it  is  the  value 
of  X  obtained  from  the  equation  when  y  is  given  the  value  zero. 
It  is  called  the  X-intercept  of  the  locus.  The  equation 
7  =  mz  -|-  b  is  called  the  common  equation  of  the  straight  line. 


X' 


1 

K 

a        1 

2 

r 

s  / 

1 

B 

1  / 

T—r 

A  / 

0 

-1 

:          3 

1 

1 

' 

/ 

2    / 

-9. 

L 

/ 

-S 

1 

J 

y' 

-4 

X 


Fig.  30. — Intercepts.  MN  is  the  line  j/  =  3x;  UK  is  the  line  y  = 
3s  +  2 ;  OB,  the  F-intercept  of  HK,  is  equal  to  2 ;  OA,  the  Z-intercept, 
is  equal  to  —2/3. 


IllustraMon  1.     Sketch  the  line  y  =  2s  +  3. 

This  line  is  3  units  higher  than  the  line  y  =  2x.  Hence  through  the 
point  (0,  3)  on  the  7-axis,  draw  a  line  of  slope  2,  which  is  the  required 
line.  I 

Illustration  2.    Sketch  the  line  ?/  =  —  2.r  —  1. 


§16]  RECTANGULAR  COORDINATES  41 

The  line  is  1  unit  lower  than  the  line  y  =  —2x.  Hence  through 
the  point  (0,  —1)  on  the  F-axis,  draw  a  line  of  slope  (—2).  The  line 
lies  halfway  between  lines  C  and  D  (Fig.  29). 

Illustration  3.     Draw  the  line  whose  equation  is  4a!  —  2?/  —  3  =  0. 

Solve  the  equation  for  y  by  transposing  the  terms  4a;  and  (—3)  to 
the  right  member  and  dividing  both  members  by  (  —  2),  then 

y  =  2x-i. 

Hence  through  the  point  (0,  — f )  on  the  y-axis  draw  a  line  of  slope  2. 

Exercises 

1.  Sketch,  from  inspection  of  the  equations,  the  lines  given  by: 

(a)  y  ^  X.  (d)  y  =  X  +  3. 

(b)  y  =  X  +  1.  (e)  y  =  X  -  1. 

(c)  y  =x  +  2.  if)  y  =x  -2. 

2.  Sketch,  from  inspection  of  the  equations,  the  lines  given  by: 

(a)  y  =  ix.  (/)  y  =  -\x. 

(6)  y  =  ix.  (g)  y  =  -x. 

(c)  y  =  X.  (h)  2/  =  -2a;. 

{d)  y  =  2x.  (i)  y  =  -3x. 

-     (e)  2/  =  3a;.  (j)  y  =  y/2  x. 

3.  Sketch  the  lines  given  by  the  equations: 

(a)  a;  =  3.  {d)  y  =  1.  (s)  2/  =  0. 

(6)  a;  =  6.  (e)  y  =  5.  (A)  x  =  0. 

(c)  X 2.  t/)  y  =  -3.  (i)a;2  =  4. 

4.  Sketch  and  name  the  slope  and  F-intercept  in  each  of  the 
following: 

(o)  2/  =  a;  +  1.  (/)  2/  =  3x  +  4. 

(6)  2/  =  ia;  +  1.  (?)  2/  =  a;  -  6. 

(c)  2/  =  -2a;  +  4.  Qi)  2/  =  fx  +  8. 

{d)  2/  =  6x  +  3.  (i)  2/  =  -3x  +  4. 

(e)  y  =  —Sx  —  2.  0')  2/  =  —  ia;  —  3. 

5.  Give  the  slope  and  F-intercept  for  each  of  the  following : 

(a)  y  =2x  +  Z.  (/)  3y  -  6x  =  12. 

(6)  y  =  3x  -2.  (9)  y  +x  =  1. 

(c)  2/  =  -3x  -  1.  I^h)  3y'+  2x  =  7. 

(d)  2/  =  5x—  6.  (i)  X  —  ^  =  6. 

(e)  22/  =  X  +  4.  (j)  X  -  22/  =  I. 


42  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§16 

6.  Find  the  X-intercept  and  the  }'-intercept  for  each  of  the 
following : 

(.a)  Sx  -  2y  =  5.  (e)  y  -  2x  ~  6  ,=  0. 

(6)  2x  +  y  =Q.  if)  2y  +  3x  +  5  =  0. 

(c)  X  -y  =  7.  (g)  X  +  y  +  I  =  0. 

(d)  y  -3x  =  5.  (h)  5y  -  3x  +  10  =  0. 

7.  Name  the  slope  and  the  F-intercept  in  each  of  the  following; 

(a)  2y  =x  +  4:.  {f)  ix  =  3y  -  6. 

(6)  y  -2x-3  =0.  (ff)  Wx-y  =  7. 

(c)  y  +  fa;  +  J  =  0.  (h)  ax  -\=y. 

(d)  2y  -\-3x  =  4.  (i)  ax  +  by  =  c. 

(e)  2x  -3y  =  6.  0')  x/a  +  y/b  =  1. 

8.  What  is  the  equation  of  the  X-axis?     Of  the  K-axis? 

9.  What  ia  the  equation  of  a  line  parallel  to  the  X-axis  4  units 
above?     3  units  above?     10  units  below?     60  units  below? 

10.  What  is  the  equation  of  a  line  parallel  to  the  F-axis  3  units 
to  the  right?  20  units  to  the  right?  7  units  to  the  left?  100  units 
to  the  left? 

11.  Plot  the  following  pairs  of  points  on  squared  paper,  and  draw 
the  line  determined  by  each  pair: 

(a)  (-1,  3)  and  (5,  -6) 
(6)  (-2,  -5)  and  (3,  4) 
(c)    (1,  1)  and  (7,  -8). 

Find  the  slope  and,  by  means  of  similar  triangles,  find  the  F-intercept 
of  each  line.  Write  the  equation  of  each  line  by  replacing  m  and  6 
in  y  =  mx  -|-  6  by  the  values  found  for  slope  and  intercept.  Test 
the  correctness  of  the  equations  by  substituting  for  x  and  y  the  co- 
ordinates of  the  given  points. 

12.  A  head  of  100  feet  of  water  causes  a  pressure  at  the  bottom  of 
43.4  pounds  per  square  inch.  Draw  a  graph  showing  the  relation 
between  head  and  pressure,  for  all  heads  of  water  from  0  to  200  feet. 

StTGGESTiON:  There  are  several  ways  of  proceeding.  Let  pounds 
per  square  inch  be  represented  by  abscissas  or  x,  and  feet  of  water  be 
represented  by  ordinates  or  y.  Since  negative  numbers  are  not  in- 
volved ia  this  exercise,  the  origin  may  be  taken  at  or  near  the  lower 
left  corner  of  the  squared  paper.  Draw  a  line  through  the  points 
(0,  0)  and  (86.8,  200).     This  will  be  the  required  graph.     Otherwise 

100 
produce  the  equation  y  =  73-72;  from  the  proportion  x:y=  43.4 :  100 


§16]  RECTANGULAR  COORDINATES  43 

and  then  draw  the  graph  from  the  fact  that  the  latitude  is  always 

TK~r  of  the  longitude.     In  drawing  this  graph  let  2  centimeters  on  the 

X-axis  represent  10  units,  and  1  centimeter  on  the  F-axis  represent 
10  units.  Be  sure  that  the  scales  are  numbered  and  labelled  in 
accordance  with  suggestions  (4),  (5),  and  (6)  of  §12.  On  the  X-axis 
mafk  only  the  points  corresponding  to  hundreds  of  pounds,  and  on' 
the  y-axis  mark  only  the  points  corresponding  to  tens  of  feet. 

13.  From  the  straight  line  drawn  in  exercise  12,  find  pressure  meas- 
ured in  pounds  corresponding  to  13.1,  112.6,  93.7,  and  187.5  feet  of 
water. 

14.  From  the  straight  line  drawn  in  exercise  12,  find  the  head  in 
feet  of  water  corresponding  to  1123,  178,  and  89  pounds  per  square 
inch. 

16.  A  pressure  of  1  pound  per  square  inch  is  equivalent  to  a  column 
of  2:04  inches  of  mercury,  or  to  one  of  2.30  feet  of  water.  Draw  a 
graph  showing  the  relation  between  pressure  expressed  in  feet  of  water 
and  pressure  expressed  in  inches  of  mercury. 

StrGGBSTiON:  Let  x  =  inches  of  mercury  and  y  =  feet  of  water. 
First  properly  number  and  label  the  X-axis  to  express  inches  of  mer- 
cury and  number  and  label  the  K-axis  to  express  feet  of  water.  Since 
negative  numbers  are  not  involved  in  this  exercise,  the  origin  may  be 
taken  at  the  lower  left-hand  corner  of  the  squared  paper.  First  locate 
the  point  x  =  2.04;  y  =  2.30  (which  are  the  corresponding  values 
given  by  the  problem)  and  draw  a  line  through  it  and  the  origin.  This 
is  the  required  locus  since  at  all  points  we  must  have  the  proportion 
x:y::  2.04 :  2.30,  which  says  that  the  ordinate  of  every  point  of 
the  locus  is  2.30/2.04  times  the  abscissa  of  that  point. 

16.  A  certain  mixture  of  concrete  (in  fact,  the  mixture  1:2:5)  con- 
tains 1.4  barrels  of  cement  in  a  cubic  yard  of  concrete.  Draw  a 
graph  showing  the  cost  of  cement  per  cubic  yard  of  concrete  for  a 
range  of  prices  of  cement  from  $0.80  to  $2.00  per  barrel. 

Suggestion:  Let  x  be  the  price  per  barrel  of  cement  and  y  be  the 
cost  of  the  cement  in  1  cubic  yard  of  concrete.  Let  2  centimeters  on 
both  vertical  and  horizontal  scales  represent  10  cents.  Number  only 
the  points  representing  multiples  of  10  cents.  Since  that  portion  of 
the  graph  near  the  origin,  namely  to  the  left  of  0,.80  and  below  1.12 
will  not  be  used,  place  the  scales  on  the  horizontal  and  vertical  lines 
passing  through  the  point  (0.80,  1.00)  and  place  this  point  at  or  near 
the  lower  left  corner  of  the  paper.  The  X-  and  F-axes  will  not 
appear  on  the  drawing. 

17.  Draw  a  graph  showing  the  cost  per  cubic  yard  of  concrete  for 


44  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§17 

various  prices  of  cement,  provided  $2.10  per  yard  must  be  added  to 
the  results  of  example  16  to  cover  cost  of  sand  and  crushed  stone. 

18.  Cast  iron  pipe,  class  A  (manufactured  for  heads  under  100  feet), 
weighs,  per  foot  of  length:  4-inch,  20.0  pounds;  6-inch,  30.8  pounds; 
8-inch,  42.9  pounds.  Upon  a  single  sheet  of  squared  paper,  construct 
a  graph  for  each  size  of  pipe,  showing  the  cost  per  foot  for  all  variations 
in  market  price  between  $20  and  $40  per  ton. 

Suggestion:  If  the  horizontal  scale  be  selected  to  represent 
•price  per  ton,  the  scale  may  begin  at  20  and  end  at  40,  as  this  covers 
the  range  required  by  the  problem.  Therefore  let  1  centimeter 
represent  $1.00.  Since  the  range  of  prices  is  from  1  cent  to  2  cents 
per  pound,  the  cost  per  foot  will  range  from  20  cents  to  40  cents  for 
4-inch  pipe  and  from  42.9  cents  to  85.8  cents  for  8-inch  pipe. 
Hence  for  the  vertical  scale  10  cents  may  be  represented  by  2 
centimeters.  In  this  case  the  vertical  scale  may  quite  as  well  begin 
at  0  cents  instead  of  at  20  cents,  as  there  is  plenty  of  room  on'  the 
paper. 

19.  Show  that  the  shortest  distance  between  y  —  mx  and  y  =  mx  +  6 

is  not  6,  but — , 

20.  Pick  out  two  pairs  of  parallel  Unes  in  exercise  5,  above  .  Pick 
out  a  pair  of  parallel  lines  in  exercise  4,  above. 

17.  Line  with  Slope  and  One  Point,  or  with  Two  Points  Given. — 

The  equation  of  any  line  parallel  to  the  F-axis  is  of  the  form  x  =  a, 
which  is  an  equation  in  which  the  variable  y  does  not  appear.  The 
equation  of  all  other  lines  may  be  written  in  the  form 

y  =  mx  +  h,     j  (1) 

in  which  m  is  the  slope  of  the  line  and  h  is  the  F-intercept.  Two 
important  special  cases  are  explained  below. 

Illustration  1.  Find  the  equation  of  the  line  of  slope  4  which  passes 
through  the  point  (2,  3). 

Since  to  =  4,  equation  (1)  becomes 

y  =4x  +  b.  (2) 

Replacing  a;  by  2  and  y  by  3,  we  get 

3  =  8  -F  6,  or  6  =  -6. 
Hence  the  equation  of  the  line  of  slope  4  passing  through  (2,  3)  is 

y  =  4x  -  5,  (3) 


:  §17]  RECTANGULAR  COORDINATES  45 

Illustration  2.  Find  the  equation  of  the  line  passing  through  the 
pomts  (2,  3)  and  (4,  1). 

Substituting  the  given  values  of  x  and  y  in  equation  (1)  we  have 

3  =  2m  +  6 
1  =  4w  +  6. 

Solving  these  equations  for  the  two  unknown  numbers,  m  and  6,  we 
find 

6  =  5 
m  =  —1, 

so  that  the  equation  of  the  line  passing  through  the  given  points  is 

2/  =  -X  +  5. 

In  like  manner  the  equation  of  a  line  passing  through  any  two  given 
points  may  be  found.  In  geometry  we  learned  that  two  points  de- 
termine a  straight  line,  and  in  the  present  problem  the  coordinates 
of  two  given  points  are  necessary  and  sufficient  for  the  determination 
of  the  equation  of  the  line. 

Exercises 

1.  Find  the  equation  of  the  line  determined  by  each  of  the  following 
conditions: 

(a)  Passes  through  (2,  5)  and  has  slope  3. 

(6)  Passes  through  (  —  2,  6)  and  has  slope  — 2^. 

(c)   Passes  through  (4,  —1)  and  has  slope  7.41. 

2.  Find  the  equation  of  the  line  determined  by  each  of  the  following 
pairs  of  points: 

(o)  (3,  2)  and  (1,  5)  (c)   (4,  6)  and  (3,  -2) 

(6)   (1,  2)  and  (-  2,  6)  (d)  (0,  0)  and  (-2,  -3) 

3.  Show  that  the  equation  of  the  line  passing  through  (a,  0)  and 
(0,  6)  niay  be  written  in  the  form 

-+!  =  !■ 

a      0 

4.  Find  the  equations  of  the  three  sides  of  the  triangle  whose  ver- 
tices are  the  points  (0,  3),  (2,  4),  and  (5,  9). 

In  each  of  the  following  exercises  certain  observed  data  are  tabu- 
lated which  will  be  found  in  each  case  to  give  points  lying  on  a  straight 
line.  The  law  connecting  y  and  x  must  then  be  of  the  form 
y  =  mx  +  b. 


46  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§17 

6.  Find  the  law  connecting  y  and  x  when  the  following  correspond- 
ing values  are  given : 

X  I     10        25  54  72 


17        47         105         141 


Hint:  In  plotting,  take  the  origin  at  the  lower  left  corner  of  the 
squared  paper.  Let  2  centimeters  represent  10  units  on  the  X-axis 
and  1  centimeter  represent  10  units  on  the  K-axis.  To  find  the  slope 
divide  the  change  in  ?/  or  (141  —  17)  by  the  increase  in  a;  or  (72  —  10) 
which  gives  2.     Find  F-intercept  by  method  of  Illustration  1. 

6.  Find  the  law  connecting  x  and  y  from  the  following  data : 

X     I      12.0         15.3         17.8         19.0 


y     \      24.2        29.0        32.6        34.2 

Hint:  Take  origin  near  left  lower  corner  of  the  squared  paper  and 
let  4  centimeters  equal  10  units  on  each  axis. 

7.  L  is  the  latent  heat  of  steam  at  a  temperature  i°  C  Find  a 
simple  formula  giving  L  in  terms  of  t. 

«     I      75  90  100         115         125 


L     I     554        544        536        526         519 

Hint:  Call  the  lower  left  corner  of  the  squared  paper  the  point 
t  =  75,  L  =  500.    Let  1  centimeter  =  5  units  on  each  axis. 

8.  V  is  the  volume  in  cubic  centimeters  of  a  certain  weight  of  gas 
at  temperature  t°  C,  the  pressure  being  constant.  Find  the  law 
connecting  V  and  t. 

t     I      27.0  33.0  40.0  55.0         68.0 

V     I     109.9         112.0         114.7         120.1         125 

Hint:  Call  the  lower  left  corner  of  the  squared  paper  the  point 
t  =  25,  V  =  100.  Let  2  centimeters  equal  5  units  on  the  t  scale  and 
1  centimeter  equal  1  unit  on  the  V  scale. 

9.  I  feet  is  the  length  of  an  iron  bar  under  a  pulling  stress  of  W  tons. 
Find  the  law  connecting  I  and  W. 

W  I     0 1 1^8 3^2 4^2 6.0 

I    I     10         10.005         10.010         10.0175         10.0225         10.0325 

Hint:  Call  the  lower  left  comer  of  the  squared  paper  the  point 
TF  =  0,  Z'=  10.  Let  2  centimeters  =  1  unit  on  the  W  scale  and  1 
qentimpter  =  0.005  unit?  on  the  I  scale. 


§17]  RECTANGULAR  COORDINATES  47 

10.  The  following  table  gives  the  draw-bar  pull  in  pounds  (P)  of 
an  electric  locomotive  in  terms  of  the  current  consumed  (A).  Find 
an  approximately  correct  algebraic  formula  giving  A  for  any  value 
of  P.     Find  the  current  required  for  a  pull  of  2000  pounds. 

P     I     400        800        1370         1600        2080        2400         , 
A     I      65  86  106  116         137  150 

Hint:  Call  the  lower  left  corner  of  the  squared  paper  A  =  50,  P  = 
400.  Let  2  centimeters  equal  10  units  on  the  A  scale  and  1  centimeter 
equal  100  units  on  the  P  scale.  / 

Exercises  5-10  above  are  taken  from  Saxelby's  "Practical  Mathe- 
matics," Longmans,  Green  and  Co.,  New  York,  1905. 


CHAPTER  III 
THE  POWER  FUNCTION 

18.  Definition  of  the  Power  Function.  The  algebraic  function 
consisting  of  a  single  power  of  the  variable,  such  for  example  as 
the  functions  x'^,  x',  1/x,  1/x'',  a;^''',  etc.,  stand  next  to  the  linear 
function  of  a  single  variable,  mx  +  b,  in  fundamental  impor- 
tance.   The  function  a;"  is  known  as  the  power  function  of  x. 

19.  The  Graph  of  x^.  The  variable  part  of  many  functions  of 
practical  importance  is  the  square  of  a  given  variable.  Thus  the 
area  of  a  circle  depends  upon  the  square  of  the  radius;  the  distance 
traversed  by  a  falling  body  depends  upon  the  square  of  the 
elapsed  time;  the  pressure  upon  a  flat  surface  exposed  directly 
to  the  wind  depends  upon  the  square  of  the  velocity  of  the 
wind;  the  heat  generated  in  an  electric  current  in  a  given  time 
depends  upon  the  square  of  the  number  of  amperes  of  current, 
etc.  Each  of  these  relations  is  expressed  by  an  equation  of  the 
form  y  =  ax^,  in  which  x  stands  for  the  number  of  units  in  one  of 
the  variable  quantities  (radius  of  the  circle,  time  of  fall,  velocity  of 
the  wind,  amperes  of  current,  respectively,  in  the  above  named 
cases)  and  in  which  y  stands  for  the  other  variable  dependent 
upon  these.  The  number  a  is  a  constant  which  has  a  value 
suitable  to  each  particular  problem,  but  in  general  is  not  the  same 
constant  in  different  problems.  Thus,  if  y  be  taken  as  the  area  of 
a  circle,  y  =  irx^,  in  which  x  is  the  radius  measured  in  feet  or 
inches,  etc.,  and  y  is  measured  in  square  feet  or  square  inches, 
etc. ;  or  if  s  is  the  distance  in  feet  traversed  by  a  falling  body, 
thens  =  16.  li'',  where  i  stands  for  the  elapsed  time  in  seconds. 
In  one  case  the  value  of  the  constant  a  is  3.1416  and  in  the  other 
its  value  is  16.1. 

Let  us  first  graph  the  abstract  law  or  equation  y  =  x^,  in  which 
a  concrete  meaning  is  not  assumed  for  the  variables  x  and  y  but 

48 


§20] 


THE  POWER  FUNCTION 


49 


in  which  both  are  thought  of  as  abstract  variables, 
suitable  table  of  values  for  x  and  x^  as  follows: 


First  form  a 


X 

a;2  or  y 

X 

x^  or  y 

-  3.0 

9.00 

0.2 

0.04 

-  2.5 

6.25 

0.4 

0.16 

-  2.0 

4.00 

0.6 

0.36 

-  1.8 

3.24 

0.8 

0.64 

-  1.6 

2.56 

1.0 

1.00 

-  1.4 

1.96 

1.2 

1.44 

-  1.2 

1.44 

1.4 

1.96 

-  1.0 

1.00 

1.6 

2.56 

-  0.8 

0.64 

1.8 

3.24 

-  0.6 

0.36 

2.0 

4.00 

-  0.4 

0.16 

2.5 

6.26 

-  0.2 

0.04 

3.0 

9.00 

0.0 

0.00 

Here  we  have  a  series  of  pairs  of  values  of  x  and  y  which  are  asso- 
ciated by  the  relation  y  —  x'^.  Using  the  x  of  each  pair  of  values 
as  abscissa  with  its  corresponding  y  there  can  be  located  as  many- 
points  as  there  are  pairs  of  values  in  the  table,  and  the  array  of 
points  thus  marked  may  be  connected  by  a  freely  drawn  curve. 
To  ,draw  the  curve  upon  coordinate  paper,  form  Ml,  the  origin 
may  be.  taken  near  the  lower  mid-point  of  the  sheet,  and  2 
centimeters  used  as  the  unit  of  measure  for  x  and  y.  If  the  points 
given  by  the  pairs  of  values  are  not  located  fairly  close  together, 
it  is  obvious  that  a  smooth  curve  cannot  be  satisfactorily  sketched 
between  the  points  until  intermediate  points  are  located  by  using 
intermediate  values  of  x  in  forming  the  table  of  values.  The 
student  should  think  of  the  curve  as  extending  indefinitely 
beyond  the  limits  of  the  sheet  of  paper  used;  the  entire  locus 
consists  of  the  part  actually  drawn  and  of  the  endless  portions 
that  must  be  followed  in  imagination  beyond  the  range  of  the 
paper.  If  the  graph  of  i/  =  a;^  be  folded  about  the  F-axis,  OY, 
it  will  be  noted  at  once  that  the  left  and  right  portions  of  the 
curve  will  exactly  coincide.  The  student  will  explain  the  reason 
for  this  fact. 

20.  Parabolic  Curves.    The  equations  y  =  x,  y  —  x',  y  =  xi, 
y  =  x^  should  be  graphed  by  the  student  on  a  single  sheet'of  coor- 


60  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§20 

dinate  paper,  using  2  centimeters  as  the  unit  of  measure  in  each 
case.^  Table  II  may  be  used  to  save  numerical  computation  in 
the  construction  of  the  graphs  of  these  power  functions.  As  in 
the  case  oi  y  =  x^,  a  smooth  curve  should  be  sketched  free-hand 


t 

L 

1 
1 

1 
1 
1 
I         5 

Y       I 

1 

1 

1 
1 

1 

1/' 

-\ 

|--- 

1 
1 
1 
1 

4 

^ 



/.. 

Vf 

1         ^ 

--f-- 

--\-'f 

4-- 

1         2 

--f- 

-A--- 



I\ 

1         1 

--J-- 

/    j 

X' 

— -|-- 

\L.. 

---]/- 

— I 

X 

2          1 

1          1         0 

1 

Y'      j 

1           !          2 

Fig.  31. — The  parabola  y  =  x'. 

through  the  points  located  by  means  of  the  table  of  values,  and 
intermediate  values  of  x  and  y  should  be  computed  when  doubt 
exists  in  the  mind  of  the  student  concerning  the  course  of  the 
curve  between  any  two  points. 

1  Wlieii.aquBred  paper  ruled  in  inches  is  used  instead  of  form  M 1,  one  inch  should 
b«  taken  as  th«  unit  of  measure. 


§20] 


THE  POWER  FUNCTION 
Table  II 


51 


X 

x^' 

X' 

Vi 

</x 

^% 

1/x 

1/x^ 

0.2 

0.04 

0.008 

0.447 

0.585 

0.089 

5.000 

25.000 

0.4 

0.16 

0.064 

0.632 

0.737 

0.252 

2.500 

6.250 

0.6 

0.36 

0.216 

0.775 

0.843 

0.46S 

1.667 

2.778 

0.8 

0.64 

0.512 

0.894 

0.928. 

0.715 

1.250 

1.563 

1.0 

1. 00 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.2 

1.44 

1.72S 

1.095 

1.063 

1.312 

0.8333 

0.6944 

1.4 

1.96 

2.744 

1.183 

1.119 

1.657 

0.7143 

0.5102 

1.6 

2.56 

4.096 

1.265 

1.170 

2.034 

0.6250 

0.3906 

1.8 

3.24 

5.832 

1.342 

1.216 

2.415 

0.6556 

0.3086 

2.0 

4.00 

8.000 

1.414 

1.260 

2.828 

0.5000 

0.2600 

2.^ 

4.84 

10.65 

1.483 

1.301 

3.263 

0.4646 

0.2066 

2.4 

5.76 

13.82 

1.549 

1.339 

3.717 

0.4167 

0.1736 

2.6 

6.76 

17.58 

1.612 

1.375 

4.193 

0.3846 

0.1479 

2.8 

7.84 

21.95 

1.673 

1.409 

4.685 

0.3571 

0.1276 

3.0 

9.00 

27.00 

1.732 

1.442 

5.196 

0.3333 

0.1111 

3.2 

10.24 

32.77 

1-.789 

1.474 

5.724 

0.3125 

0.0977 

3.4 

11.56 

39.30 

1.844 

1.504 

6.269 

0.2941 

0.0865 

3.6 

12.96 

46.66 

1.897 

1.533 

6.831 

0.2778 

0.0772 

3.8 

14.44 

54.87 

1.949 

1.560 

7.407 

0.2632 

0.0693 

4.0 

16.00 

64.00 

2.000 

1.587 

8.000 

0.2500 

0.0625 

4.2 

17.64 

74.09" 

2.049 

1.613 

8.608 

0.2381 

0.0567 

4.4 

19.36 

85.18 

2.098 

1.639 

9.229 

0.2273 

0.0517 

4.6 

21.16 

97.34 

2.145 

1.663 

9.866 

0.2174 

0.0473 

4.8 

23.04 

110.6 

2.191 

1.687 

10.42 

0.2083 

0.0434 

5.0 

25.00 

125.0 

2.236 

1.710 

11.18 

0.2000 

0.0400 

5.2 

27.04 
20.16 

140.6 

2.280 

1.732 

11.85 

0. 1923 

0.0370 

5.4 

157.5 

2.324 

1.754 

12.66 

0.1862 

0.0343 

5.6 

31.36 

175.6 

2.366 

1.776 

13.25 

0.1786 

0.0319 

5.8 

33.64 

195.1 

2.408 

1.797 

13.97 

0.1724 

0.0297 

6.0 

36.00 

216.0 

2.449 

1.817 

14.70 

0.1667 

0.0278 

6.2 

38.44 

238.3 

2.490 

1.837        : 

15.44 

0.1613 

0.0260 

6.4 

40.96 

262.1 

2.530 

1.867 

16.19 

0.1563 

0.0244 

6.6 

43.56 

287.5 

2.569 

1.876 

16.96 

0.1515. 

0.0230 

6.8 

46.24 

314.4 

2.608 

1.895 

17.33 

0.1471 

0.0216 

7.0 

49.00 

343.0 

2.646 

1.913 

18.52 

0.1429 

0.0204 

7.2 

51.84 

373.2 

2.683 

1.931 

19.32 

0.1389 

0.0193 

7.4 

54.76 

405.2 

2.720 

1.949 

20.13 

0.1351 

0.0183 

7.6 

57.76 

439.0 

2.757 

1.966 

20.95 

0.1316 

0.0173 

7.8 

60.84 

474.6 

2.793 

1.983 

21.79 

0.1282 

0.0164 

8.0 

64.00 

512.0 

2.828 

2.000 

22.63 

0.1260 

0.0156 

8.2 

67.24 

551.4 

2.864 

2.017 

23.48 

0.1220 

0.0149 

8.4 

70.56 

592.7 

2.898 

2.033 

24.35 

0.1190 

0.0142 

8.6 

73.96 

636.1 

2.933 

2.049 

25.22 

0.1163 

0.0135 

8.8 

77.44 

681.5 

2.966 

2.065 

26.11 

0.1136 

0.0129 

9.0 

81.00\ 

729.0 

3.000 

2.080 

27.00 

0.1111 

0.0123 

9.2 

84.64 

778.7 

3.033 

2.095 

27.91 

0.1087 

0.0118 

9.4 

88.36 

830.6 

3.066 

2.110 

28.82 

0.1064 

0.0113 

9.6 

92.16 

884.7 

3.098 

2.125 

29.74 

0.1042 

0.0109 

9.8 

96.04 

941.2 

3.130 

2.140 

30.68 

0.1020 

0.0104 

10.0 

100.00 

1000.0 

3.162 

2.154 

31.62 

0.1000 

0.0100 

All  of  the  graphs  here  considered  have  one  important  property 
in  common,  .namely,  they  all  pass  through  the  points  (0,  0)  and 
(1,  1).    It  is  obvious  that  this  property  may  be  affirmed  of  any 


52  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§21 

curve  of  the  class  y  =  a;",  if  n  is  a  positive  number.  These  curves 
are  known  collectively  as  curves  of  the  parabolic  family,  or  simply 
parabolic  curves.  The  curve  y  =  x^  is  called  the  parabola. 
2/  =  a;'  is  called  the  cubical  parabola,  y  =  xi  is  called  the  semi- 
cubical  parabola,  etc.  Curves  for  negative  values  of  n  do  not  pass 
through  the  point  (0,  0)  and  are  otherwise  quite  distinct.  They 
are  known  as  curves  of  the  hyperbolic  type,  and  will  be  discussed 
later. 

The  student  should  cut  patterns  df  the  parabola,  the  cubical 
parabola  and  the  semi-cubical  parabola  out  of  heavy  paper  for 
use  in  drawing  these  curves  when  required.  Each  pattern  should 
have  drawn  upon  it  either  the  X-  or  F-axis  and  one  of  the  unit 
lines  to  assist  in  properly  adjusting .  the  pattern  upon  squared 
paper. 

21.  Symmetry. — In  geometry  a  distinction  is  made  between  two 
kinds  of  symmetry  of  plane  figures — symmetry  with  respect  to  a 
line  and  symmetry  with  respect  to  a  point.  A  plane  figure  is 
symmetrical  with  respect  to  a  given  line  if  the  two  parts  of  the 
figure  exactly  coincide  when  folded  about  that  line.  Thus  the  let- 
ters M  and  W  are  each  symmetrical  with  respect  to  a  vertical  line 
drawn  through  the  vertex  of  the  middle  angles.  We  have  already 
noted  that  y  =  x^  is  symmetrical  with  respect  to  OY. 

A  plane  figure  is  symmetrical  with  respect  to  a  given  point  when 
the  figure  remains  unchanged  if  rotated  180°  in  its  own  plane  about 
an  axis  perpendicular  to  the  plane  at  the  given  point.  Thus  the 
letters  N  and  Z  are  each  symmetrical  with  respect  to  the  mid-point 
of  their  central  line.  The  letters  H  and  O  are  symmetrical  both 
with  respect  to  lines  and  with  respect  to  a  point.  Which  sort  of 
symmetry  is  possessed  by  the  curve  y  =  a;'?    Why? 

Another  definition  of  symmetry  with  respect  to  a  point  is  per- 
haps clearer  than  the  one  given  by  the  above  statement:  A  curve 
is  said  to  be  symmetrical  with  respect  to  a  given  point  0  when  all 
lines  drawn  through  the  given  point  and  terminated  by  the  curve 
are  bisected  at  the  point  0. 

What  kind  of  symmetry  with  respect  to  one  of  the  coordinate 
axes  or  to  the  origin  (as  the  case  may  be)  does  the  point  (2,  3)  bear 
to  the  point  (-2,  3)?  To  the  point  (-2,-3)?  To  the  point 
(2,  -3)? 


§22] 


THE  POWER  FUNCTION 


53 


Note  that  symmetry  of  the  first  kind  means  that  a  plane  figure  is 
unchanged  when  turned  180°  about  a  certain  line  in  its  plane,  and 


l-r-Y 


Fig.  32. — The  parabolas  «/  =  a;"  for  n  =  1,  2,  3,  and  4. 

that  symmetry  of  the  second  kind  means  that  a  figure  is  unchanged 
when  turned  180°  about  a  certain  line  perpendicular  to  its  plane. 
22.  The  curves  in  Figs.  31  to  34  are  sketched  from  a  limited 


54 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§22 


number  of  points  only,  but  any  number  of  additional  values  of  x 
and  y  may  be  tabulated  and  the  accuracy,  as  well  as  the  extent, 
of  the  graph  be  made  as  great  as  desired.  A  number  of  graphs  of 
power  functions  are  shown  as  they  appear  in  the  first  quadrant 


in  Figs.  34  and  38.  The  student  should  explain  how  to  draw  the 
portions  of  the  curves  lying  in  the  other  quadrants  from  the  part 
appearing  in  the  first  quadrant. 

In  the  exercises  in  this  book  to  "draw  a  curve"  means  to  con- 


THE  POWER  FUNCTION 


55 


struct  the  curve  as  accurately  as  possible  from  numerical  or  other 
data.  To  "sketch  a  curve"  means  to  produce  an  approximate  or 
less  accurate  representation  of  the  curve,  including  therein  its 
characteristic  properties,  but  without  the  use  of  extended  numer- 
ical data.  Whenever  possible,  make  use  of  the  paper  patterns 
mentioned  in  §20. 


0    0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1  1.2  1.3  1.4  IJl  1.6  1.7  1.8  1.9  2.0 

!Fig.  34. — Graph  of  the  power  function  for  n  >0  (parabolic  curves)  in 
the  first  quadrant. 


Exercises 


1.  On  coordinate  paper  draw  the  curves  y  —  x',  y  =  x^,  y  =  x^, 
y  =  x^,  using  2  centimeters  as  the  unit  of  measure.  On  the  same 
sheet  draw  the  Unes  x  =  +1,  y  =  ±1,  y  —  +x. 

2.  On  coordinate  paper  sketch  the  curves  x  =  y^,  x  =  y^,  x  =  yi, 
X  =  y^.    Compare  with  the  curves  of  Exercise  1. 


56 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§23 


3.  Sketch  and  discuss  the  curves  y  =  xi,  y  =  xi,  y  =  xi.  Can 
any  of  these  curves  be  drawn  from  patterns  made  from  the 
curves  of  Exercise  1?    Why? 

4.  Draw  the  curve  y^  =  x*.    Compare  with  the  curve  y  =  x^. 

6.  Name  in  each  case  the  quadrants  in  which  the  curves  of  Exer- 
cises 1-4  lie,  and  state  the  reasons  why  each  curve  exists  in  certain 
quadrants  and  not  in  the  other  quadrants. 

23.  Discussion  of  the  Parabolic  Curves. — Draw  the  straight 
ines  X  =  1,  X  =  —l,y=l,y=  —1  upon  the  same  sheet  upon 

which  a  number  of  para- 
■,/J;^.  :^  I  B      bolic   curves   have   been 

drawn.  These  lines  to- 
gether with  the  coordi- 
nate axes  divide  the  plane 
into  a  number  of  rect- 
angular spaces.  In  Fig. 
35  these  spaces  are  shown 
divided  into  two  sets, 
those  represented  by  the 
cross-hatching,  and  those 
shown  plain.  The  cross- 
hatched  rectangular  spaces 
'iontain  the  lines  y  =  x 
and  y  =  —x  and  also  all 
curves  of  the  parabolic 
type.  No  parabolic  curve 
enters  the  rectangular  strips 
shown  plain  in  Fig.  35. 
The  line  y  =  x  divides  the  spaces  occupied  by  the  parabolic 
curves  into  equal  portions.  Why  does  the  curve  y  =  x'  (in  the 
first  quadrant)  lie  below  this  line  in  the  interval  a;  =  0  to  a;  =  1, 
but  above  it  in  the  interval  to  the  right  of  x  =  1  ?  On  the  other 
hand,  why  does  the  curve  y  =  xi,  or  y^  =  x  (in  the  first  quad- 
rant), lie  above  the  line  2/  =  a;  in  the  interval  a;  =  0  to  a;  =  1  and 
below  y  =  xin  the  interval  to  the  right  of  x  =  1? 

One  part  of  the  parabolic  curve  y  =  x"  always  lies  in  the  first 
quadrant.  If  n  be  an  even  integer,  another  part  of  the  curve  lies 
in  which  quadrant?  If  n  be  an  odd  integer,  the  curve  lies  in  which 
quadrants? 


Fig.  35. — The  regions  of  the  parabohc 
and  the  hyperboUc  curves.  All  parabolic 
curves  he  within  the  cross-hatched 
region.  All  hj^jerboUc  curves  he  within 
the  region  shown  plain. 


§23]  THE  POWER  FUNCTION  57 

If  the  exponent  n  of  any  power  function  be  a  positive  fraction, 
say  m/r,  the  equation  of  the  curve  may  be  written  y  =  x".  If 
in  this  case  both  m  and  r  be  odd,  the  curve  lies  in  which  quadrants? 
If  m  be  even  and  r  be  odd,  the  curve  lies  in  which  quadrants? 
If  m  be  odd  and  r  be  even,  the  curve  lies  in  which  quadrants? 
If  both  m  and  r  be  even  the  curve  lies  in  which  quadrants? 

A  curve  which  is  symmetrical  to  another  curve  with  respect  to  a 
line  may  figuratively  be  spoken  of  as  the  reflection  or  image  of  the 
second  curve  in  a  mirror  represented  by  the  given  line. 

Exercises 
Exercises  1-5  refer  to  curves  in  the  first  quadrant  only. 

3 

1.  The  expressions  x',  x^,  x',  x*  are  numerically  less  than  x  for 
values  of  x  between  0  and  1.     How  is  this  fact  shown  in  Fig.  34? 

2.  The  expressions  x^,  x^,  a;',  s*  are  numerically  greater  than  x  for 
all  values  of  x  numerically  greater  than  unity.  How  is  this  fact 
pictured  in  Fig.  34? 

3.  For  values  of  x  between  0  and  1,  x*  <x^  <x''  <x^  <  x.  For 
values  X  >  1,  X*  >  x^  >  x^  >  ^  >  X.  Explain  how  each  of  these 
facts  is  expressed  by  the  curves  of  Fig.  34. 

2  i  i  1 

4.  Show  that  the  graphs  y  =  x^,  y  =  a;',  y  =  x'l  y  =  x^  are 
the  reflections  of  i/  =  x^,  y  =  x',  y  =  x\  y  =  x\  in  the  line  y  =  x. 

6.  Sketch  on  a  single  sheet  of  squared  paper  without  tabulating 
the  numerical  values,  the  following  loci:  y  =  x^",  y  =  a;"',  y  =  a;'"", 

y    =  a;0.01 

The  following  are  to  be  discussed  for  all  quadrants. 

6.  Sketch,  without  tabulating  numerical  values,  the  following  loci 
y'  =  x\  y*  =  x',  y^  =  x^,  y'  =  x^,  y^  =  x^. 

7.  Show  that  y  =  —x'  is  the  reflection  oi  y  =  x'  in  the  X-axis. 

8.  Sketch  the  following;  y  =  —x,  y  =  —x^,  y  =  —x',  y'  =  —  x^, 
y^  —  —  x^. 

9.  A  ball  roUs  down  a  smooth  inclined  plane  making  an  angle  of 
45°  with  the  horizontal.  The  distance  s  measured  in  feet  along  the 
incline  is  given  by  the  formula 

s  =  11.4*2 

where  t  is  time  in  seconds.     Draw  a,  graph  for  this  equation.    Let 
time  t  be  represented  by  distances  along  the  axis  of  abscissas  and  dis- 


58 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§24 


tance  s  along  the  axis  of  ordinates.    Let  2  centimeters  represent  one 
second  on  one  axis  and  10  feet  on  the  other. 

24.  Hyperbolic  Curves.  Loci  of  equations  of  the  form  yx"  =  1, 
OT  y  =  l/x",  where  n  is  positive,  are  called  hyperbolic  curves. 
The  fundamental  curve  xy  =  1,  or  y  =  l/x  is  called  the  rec- 
tangular hyperbola.    Its  graph  is  given  in  Fig.  36,  but  the  curve 


Fig.  36. — The  hyperbolas  y  =  x"  ioi  n  =  —  1,  —2,  and  —3. 


should  be  drawn  independently  by  the  student,  using  2  centi- 
meters as  the  unit  of  measure.  Its  relation  to  the  X-  and 
y-axes  is  most  characteristic.  For  a  very  small  positive  value  of 
X,  the  value  of  y  is  very  large,  and  as  x  approaches  0,  y  increases 
indefinitely.  But  the  function  is  not  defined  for  x  =  0,  for  the 
product  xy  cannot  equal  1  if  x  be  zero.     For  numerically  small  but 


THE  POWER  FUNCTION 


59 


negative  values  of  x,  y  is  negative  and  numerically  very  large,  and 
becomes  numerically  larger  as  x  approaches  0. 

Instead  of  saying  that  "y  increases  in  value  without  limit,"  it 
is  just  as  common  to  say  "y  becomes  infinite;"  in  fact,  "infinite" 
is  merely  the  Latin  equivalent  of  "no  limit."  It  is  often  written 
2/  =  00 .  This  is  a  mere  abbreviation  for  the  longer  expressions, 
"y  becomes  infinite"  or  "y  increases  in  value  without  limit." 
The  student  must  be  cautioned  that  the  symbol  <»  does  not  stand 


Fig.  37. — The  hyperbolas  y  =  x"  for  n 


-i,  —i,  —21  and  —J 


for  a  number,  and  that  "y  =  oo"  must  not  be  interpreted  in  the 
same  way  that  "y  =  5"  is  interpreted. 

As  X  increases  from  numerically  large  negative  values  to  0, 
y  continually  decreases  and  becomes  negatively  infinite  (abbre- 
viated 2/  =  —  00  ).  As  a;  decreases  from  numerically  large  positive 
values  to  0,  y  continually  increases  and  becomes  infinite.  Thus, 
in  the  neighborhood  oi  x  =  0,  y  is  discontinuous,  and,  in  this  case, 
the  discontinuity  is  called  an  infinite  discontinuity. 


60 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§24 


0    0.1  0.2  0.3  0.4  O.S  0.6  0.7  0.8  0.9  1.0  1.1  1.2  1.3  1.1  1.6  1.6  1.7  1.8  1.9  2.0 

Fig.  38. — Hsrperbolic  curves  in  the  first  quadrant.       y  =  l/s^-^o'is 
the  adiabatic  curve  for  air. 

On  account  of  the  symmetry  in  xy  =  l,]i  we  look  upon  x  as  a 
function  o^  y,  all  of  the  above  statements  may  be  repeated,  merely 

interchanging  x  and  y  wherever 
they  occur.  Thus,  there  is  an  in- 
finite discontinuity  in  x,  as  y 
passes  through  the  value  0. 

The  lines  XX'  and  YY'  which 
these  curves  approach  as  near  as' 
we  please,  but  never  meet,  are 
called  the  asymptotes  of  the 
hyperbola. 

All  other  curves  of  the  hyper- 
bolic   family,    such    as    yx"^  =  1, 
^2/"  =  1,  2/'a;'  =  1,  y^x*  =  1    and 
,  the   like,   approach   the   X-   and 

y-axes  as  asymptotes.    The  rates  at  which  they  approach  the 


^\ 

^=^^ 

^^m ^^ 

1 

wMmm 

~\ 

^ 

=^= — 1 

Fig.  39. — A  hyperbola 
formed  by  capillary  action  of 
two  converging  plane  plates. 


§25]  THE  POWER  FUNCTION  61 

axes  depends  upon  the  relative  magnitudes  of  the  exponents  of  the 
powers  of  x  and  y;  the  quadrants  in  which  the  branches  lie  depend 
upon  the  oddness  or  evenness  of  these  exponents. 

Exercises 

1.  Draw  accurately  upon  squared  paper  the  loci,  xy  =  1,  xy'  =  1, 
x'y  =  1.     Use  2  centimeters  as  unit  and  make  a  pattern  for  xy  =  1. 

2.  Show  that  the  curves  of  the  hyperboUc  type  lie  in  the  rectangu- 
lar regions  shown  plain,  or  not  cross-hatched,  in  Fig.  35. 

3.  In  what  quadrants  do  the  branches  of  x^^y''  =  1  he? 

4.  How  does  the  locus  of  x^y^  —  1  differ  from  that  oixy  =  1  ? 

6.  Sketch,  showing  the  essential  character  of  each  locus,  the  curves 
x^y  =  1,  x^y  =  1,  s"""?/  =  1. 

6.  Show  that  xy  =  d  passes  through  the  point  (\/a,  -y/a) ',  that  xy 
=  a'  passes  through  (a,  a)  and  can  be  made  from  xy  =  Ihy  "stretch- 
ing" (if  a  >  1)  both  abscissas  and  ordinates  of  xy  =  1  in  the  ratio 
l:a.i 

25.  Symmetry.  Some  of  the  facts  of  symmetry  respecting  two 
portions  of  the  same  parabola  or  hyperbola  may  be  readily  ex- 
tended by  the  student  to  other  curves.  First  answer  the  following 
questions : 

How  are  the  points  (a,  6)  and  (  —  a,  b)  related  to  the  F-axis? 

How  are  the  points  (a,  6)  and  (a,  —6)  related  to  the  X-axis? 

How  are  the  points  (a,  b)  and  (6,  a)  related  to  the  line  y  =  xl 
Prove  the  result  by  plane  geometry. 

The  following  may  then  be  readily  proved  by  the  student: 

Theorems  on  Loci 

I.  7/  X  he  replaced  by  (—x)  in  any  eqvution  containing  x  and  y, 
the  new  graph  is  the  reflection  of  the  former  in  the  axis  YY'. 

II.  If  y  be  replaced  by  (—y)  in  any  equation  containing  x  and  y, 
the  new  graph  is  the  reflection  of  the  former  in  the  axis  XX'. 

III.  If  x  and  y  be  interchanged  in  any  equation  containing  x 
and  y,  the  new  graph  is  the  reflection  of  the  former  in  the  line  y  =  x. 

IV.  If  an  equation  remains  unchanged  when  x  is  replaced  by 
{—x),  its  graph  is  symmetrical  with  respect  to  the  Y-axis. 

1  To  "elongate"  or  "stretch"  in  the  .ratio  2  :3  naeans  to  change  tfie  length  of  a 
line  segment  so  that  (original  length)  :  (new  or  stretched  length)  =  2:3. 


62 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§26 


v.  If  an  equation  remains  unchanged  when  y  is  replaced  by  {—y), 
its  graph  is  symmetrical  with  respect  to  the  X-axis. 

VI.  If  an  equation  remains  unchanged  when  x  and  y  are  inter- 
changed, its  graph  is  symmetrical  with  respect  to  the  line  y  =  x. 

VII.  If  an  equation  remains  unchanged  when  x  is  replaced  by 
i—x)  and  y  by  {—y),  its  graph  is  symmetrical  with  respect  to  the 
origin. 

VIII.  If  an  equation  remains  unchanged  when  x  is  replaced  by 
(—J/)  and  y  is  replaced  by  (.—x),  its  graph  is  symmetrical  with  re- 
spect to  the  line  y  =  —x. 

26.  The  Variation  of  the  Power  Function.  The  symmetry  of 
the  graphs  of  the  power  function  with  respect  to  certain  lines  and 
points,  while  of  interest  geometrically,  nevertheless  does  not  con- 
stitute the  most  important  fact  in  connection  with  these  functions. 
Of  more  importance  is  the  law  of  change  of  value  or  the  law  by  which 
the  function  varies.  Thus  returning  to  a  table  of  values  for  the 
power  function  x^  for  the  first  quadrant. 


X 

xi 

d 

0 

0 

1 

1 

1 

4 

? 

4, 

a 

1 

\ 

4 

5. 

3 

g 

4 

1 

% 

7 

2 

¥- 

5 

¥ 

I 

3 

¥ 

i 

-V- 

5  . 

4 

¥- 

i 

we  note  that  as  x  changes  from  0  to  ^  the  function  grows  by  the 
small  amount  \.  As  x  changes  from  \  by  another  increment  of 
\  to  the  value  1,  the  function  increases  by  f  to  the  value  1. 
As  X  grows  by  successive  steps  or  increments  of'|  unit  each,  it 
is  seen  that  x"^  grows  by  increasingly  greater  and  greater  steps, 
until  finally  the  change  in  x"^  produced  by  a  small  change  in  x 
becomes  very  large.  Thus  the  step  by  step  increase  in  the  func- 
tion is  a  rapidly  augmenting  one,  as  is  shown  by  the  column  of 


§27]  THE  POWER  FUNCTION  63 

differences  headed  "d"  in  the  table.  Even  more  rapidly  does  the 
function  x^  gain  in  value  as  x  grows  in  value.  On  the  contrary, 
for  positive  values  of  x  the  power  functions  1/x,  1/x^,  1/x^,  etc., 
decrease  in  value  as  x  grows  in  value.  Referring  to  the  definition 
of  the  slope  of  a  curve  given  in  §15,  we  see  that  the  parabolic 
curves  have  a  positive  slope  in  the  first  quadrant,  while  the  hyper- 
bolic curves  have  always  a  negative  slope  in  the  first  quadrant. 

The  law  of  the  power  function  is  stated  in  more  definite  terms 
in  §34.  That  section  may  be  read  at  once,  and  then  studied 
again  in  connection  with  the  practical  work  which  precedes  it. 

27.  Increasing  and  Decreasing  Functions. — As  a  point  passes 
from  left  to  right  along  the  X-axis,  x  increases  algebraically. 
As  a  point  moves  up  on  the  F-axis,  y  increases  algebraically  and 
as  it  moves  down  on  the  F-axis,  y  decreases  algebraically.  An 
increasing  function  of  x  is  one  such  that  as  x  increases  algebraic- 
ally, y,  or  the  function,  also  increases  algebraically.  By  a 
decreasing  function  of  x  is  meant  one  such  that  as  x  increases 
algebraically,  y  decreases  algebraically.  Graphically,  an  increas- 
ing function  is  indicated  by  a  rising  curve  as  a  point  moves  along 
it  from  left  to  right.  The  power  function  y  =  s^  {n  positive)  is  an 
increasing  function  of  x  in  the  first  quadrant  and  y  =  x~^  (— n 
negative)  is  a  decreasing  function  of  x  in  the  first  quadrant. 

The  power  function  y  =  x^  \&  an  increasing  function  for  all 
positive  atid  for  all  negative  values  of  x,  while  y  =  x'^\&&  decreasing 
function  in  the  second  quadrant  but  an  increasing  function  in  the 
first  quadrant.  In  a  case  like  y  =  +xi,  where  y  has  two  values 
for  each  positive  value  of  x,  it  is  seen  that  one  of  these  values 
increases  with  x  while  the  other  decreases  with  x. 

Exercises 

1.  Consider  the  function  y  =  +x^.  As  x  grows  by  successive  steps 
of  one  unit  each,  does  the  function  grow  by  increasingly  greater  and 
greater  steps  or  not?  Is  the  slope  of  the  curve  an  increasing  or  a 
decreasing  function  of  x? 

2.  Does  the  algebraic  value  of  the  slope  oi  xy  =  1  increase  with  x  in 
the  first  quadrant? 

3.  As  £  changes  from  —5  to  +5  does  the  slope  oi  y  =  x^  always 
increase  algebraically? 


64  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§28 


4.  Express  in  the  language  of  mathematics  the  fact  that  the  curves 
y  =  I",  when  n  is  a  rational  number  greater  than  unity,  are  concave 
upward. 

Answer:  "When  n  is  greater  than  unity,  the  slope  of  the  curve 
increases  as  x  increases." 

Express  in  a  similar  way  the  fact  that  the  curves  y  =  x^/"  are 
concave  downward. 

28.  The  Graph  of  the  Power  Function  when  x»  has  a  Coeffi- 
cient.    If  numerical  tables  be  prepared  for  the  equations 

y  =  x^ 
y'  =  2x' 


and 


0\         1  X>  2      -4      -3       -2       -1      O 

(a)  (b) 

Fig.  40. — (o)  The  curves  y  =  x'  and  y'  =  2x'.     (b)  The  curves  y  = 

x^  and  2/  =  (I)   • 

then  for  like  values  of  x  each  ordinate  of  the  second  curve  will 

be  two  times  the  corresponding  ordinate  of  the  first  curve.    These 

curves  are  shown  in  Fig.  40a.    For  each  position  of  P  on  the 

curve  y'  =  2x\  DP  =  2DQ. 

It  is  obvious  that  the  curve 

J  i-u  V   =  X"  (1) 

and  the  curve  y,  ^  a^„  ^2) 

are  similarly  related;  the  ordinate  of  any  point  of  the  second  graph 
can  be  made  from  the  corresponding  ordinate  (i.e.,  the  ordinate 
having  the  same  abscissa)  of  the  first  graph  by  multiplying  the 
former  by  a.    If  o  be  positive  and  greater  than  unity,  this  corre- 


§28]  THE  POWER  FUNCTION  65 

sponds  to  stretching  or  elongating  all  ordinates  of  (1)  in  the  ratio 
1 :  o;  if  a  be  positive  and  less  than  unity,  it  corresponds  to  con- 
tracting or  shortening  all  ordinates  of  (1)  in  the  ratio  1 :  a. 

For  example,  the  graph  of  y'  =  ax'  can  be  made  from  the 
graph  of  y  =  x"ii  the  latter  be  first  drawn  upon  sheet  rubber,  and  if 
then  the  sheet  be  uniformly  stretched  in  the  y  direction  in  the  ratio 
1 :  a.  If  the  curve  be  drawn  upon  sheet  rubber  which  is  already 
under  tension  in  the  y  direction  and  if  the  rubber  be  allowed  to 
contract  in  the  y  direction,  the  resulting  curve  has  the  equation 
y  =  ax"  where  a  is  a  proper  fraction  or  a  positive  number  less  than 
unity. 

The  above  results  are  best  kept  in  mind  when  expressed  in  a 
slightly  different  form.  The  equation  y'  =  a-x"  can,  of  course,  be 
written  in  the  form  (y'/a)  =  a;".  Comparing  this  with  the  equa- 
tion y  =  x",  we  note  that  (y'/a)  =  y  or  y'  =  ay,  therefore  we  may 
conclude  generally  that  substituting  (y'/a)  for  y  in  the  equation  of 
any  curve  multiplies  all  of  the  ordinates  of  the  curve  by  a.  For 
example,  after  substituting  (y'/2)  for  y  in  any  equation,  the  new 
ordinate  y'  must  be  twice  as  large  as  the  old  ordinate  y,  in  order 
that  the  equation  remain  true  for  the  same  value  of  x. 

(x'\  ** 
—  I  , 

that  is,  substituting  (  — )  for  x  in  any  equation  multiplies  all  of  the 

abscissas  of  the  curve  by  a.  See  Fig.  406.  Multiplying  all 
of  the  abscissas  of  a  curve  by  a  elongates  or  stretches  all  of  the 
abscissas  in  the  ratio  ^  1  :  a  if  a  >  1,  but  contracts  or  shortens  all 
of  the  abscissas  if  o  <  1.  As  the  above  reasoning  is  true  for  the 
equation  of  any  locus,  we  may  state  the  results  more  generally 
as  follows: 

Theorems  on  Loci 

IX.  Substituting  ( -)  for  z  in  the  equation  of  any  locus  multi- 
plies all  of  the  abscissas  of  the  curve  by  a. 

X.  Substituting  I  - )  for  y  in  the  equation  of  any  locus  multiplies  all 
of  the  ordinates  of  the  curve  by  a. 

1  See  footnote,  p.  61. 


66  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§29 

Note:  It  is  not  necessary  to  retain  the  symbols  x'  and  y'  to 
indicate  new  variables,  if  the  change  in  the  variable  be  otherwise 
understood. 

Exercises 

1.  Without  actual  construction,  compare  the  graphs  y,  =  a;^  and 

^2  1  2 

y  =  5x2;  J/  =  a;2  and  ^  =  "2  ;  2/  =  -  and  J/  =  -;  y  =  *»  and  y  =  2x^; 

s 
y  =  x^  and  2/  =  -g  ' 

2.  Without  actual  construction,  compare  the  graphs  y  =  x'  and 
2/  =  f|V;  2/  =  s' and ^=  x^;y  =  x^a.ndy  =  {^j;  y  =  x^  and  |  =  x\ 

3.  Compare  y^  =  a;=  and  i/^  =  \k]   ;  j/^  =  i'  and  (gj    =  x\-  y'  =x' 

4.  Compare   the   curves  t/    =    2x^  and  2/    =    2  ( s  j    ;  3?/''  =  a;'  and 

32/==  (l)  ';  2/^  =  X'  and  (2)/)^'  =  {Zxy. 

5.  Compare  ?/  =    a;   +   3  and   y  =  2  {x  +  3);  y  =  2x  —  1  and 

I  =2a;  -  1;  2/  =  2a;  -  1  and  22/  =  2a;  -  1. 

The  following  exercise  involves  a  different  principle  from  that  used 
above,  which  the  student  should  reason  out  for  himself. 

6.  Without  actually  constructing  the  curves,  compare  the  curves, 
for  2/  =  2a;  +  3  and  y  =  2x  +  5;  y  =  x^  and  y  =  x^  +  l;  y  ==  x'  and 
2/  =  a;'  +  2;  2/  =  a;'  and  y  =  x'  —  1;  y  =  x'  and  y  =  x'  +  i; 
y  =~  and  y  =  — \-  2;y  =  x^  and  2/  =  (x  —  1)^. 

29.  Change  of  Unit.  To  produce  the  graph  of  2/  =  lOa;^  from 
that  of  2/  =^  a;^,  the  stretching  of  the  ordinates  in  the  ratio  1 :  10 
need  not  actually  be  performed.  If  the  unit  of  the  vertical  scale 
of  2/  =  a;^  be  taken  1/10  of  that  of  the  horizontal  scale,  and  the 
proper  numerical  values  be  placed  upon  the  divisions  of  the 
scales,  then  obviously  the  graph  ol  y  =  x^  may  be  used  for  the 
graph  of  2/  =  lOx''.  Suitable  change  in  the  unit  of  measure  on  one 
or  both  of  the  scales  of  2/  =  a?"  is  often  a  very  desirable  method  of 
representing  the  more  general  curve  y  —  ax^. 

An  interesting  example  is  given  in  Fig.  41.    The  period  of  vi- 


THE  POWER  FUNCTION 


67 


1.6 

1.4 

■1.2 

^ 

"^ 

-1.0 

|0.8 

|o.6 

0.4 

0  2 

^ 

^ 

y 

y 

/ 

/ 

/ 

/ 

0  20  40  60  80  100  120  140  160 180  200 
liength  in  Ooi. 

Fig.    41.^ — Relation  of  length  of  a, 
simple  pendulum  to  period  of  vibration. 


bration  of  a  simple  pendulum  is  given  by  the  formula  T  =  Tr-\/l/g. 
When  g' =  981  cm.  per  second  per  second  (abbreviated  cm. /sec. 2) 
this  gives  T  =  O.W03-\/l,  which  for  many  purposes  is  sufficiently 
accurate  when  written  T  =  0.10\/l.  In  this  equation  T  must 
be  in  seconds  and  I  in  centimeters.  Thus  when  I  =  100  cm.,  T 
=  1  sec,  so  that  the  graph  may  be  made  by  drawing  the  parabola 
y  =  ^/x  from  the.  pattern 
previously  made  and  then 
attaching  the  proper  num- 
bers to  the  scales,  as  shown 
in  Fig.  41. 

30.  Variation.  The  re- 
lation between  y  and  a;  ex- 
pressed by  the  equation 
y  =  ax",  where  n  is  any 
positive  number,  is  often 
expressed  by  the  state- 
ment "y  varies  as  the  nth 
power  of  X,"  or  by  the 
statement  "y  is  proportional  to  X"."  Likewise,  the  relation 
y  =  a/x",  where  n  is  positive,  is  expressed  by  the  statement 
"y  varies  inversely  as  the  nth  power  of  x."  The  statement  "the 
elongation  of  a  coil  spring  is  proportional  to  the  weight  of  the 
suspended  mass"  tells  us 

y  =  mx  (1) 

where  y  is  the  elongation  (or  increase  in  length  from  the  natural 
or  unloaded  length)  of  the  spring,  and  x  is  the  weight  suspended  by 
the  spring,  hut  it  does  not  give  us  the  value  of  m.  The  value  of  m 
may  readily  be  determined  if  the  elongation  corresponding  to  a 
given  weight  be  given.  Thus  if  a  weight  of  10  pounds  when  sus- 
pended from  the  spring  produces  an  elongation  of  2  inches  in  the 
length  of  the  coil,  then,  substituting  x  =  10  and  y  =  2  m  (1), 

and  hence  2  =  mlO 

m  =    . 

If  this  spring  be  used  in  the  construction  of  a  spring  balance,  the 
length  of  a  division  of  the  uniform  scale  corresponding  to  1  pound 
will  be  1/5  inch. 


68  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§30 

A  special  symbol,  « ,  is  often  used  to  express  variation.    Thus 

states  that  y  varies  inversely  as  d^.    It  is  equally  well  expressed  by 

k 

where  A;  is  a  constant  called  the  proportionality  factor. 

The  statements  "y  varies  jointly  as  u  and  v,"  and  "y  varies 
directly  as  u  and  inversely  as  v,''  mean,  respectively, 

y  =  kuv 
hu 

Thus  the  area  of  a  rectangle  varies  jointly  as  its  length  and 
breadth,  or 

A  =  kLB. 

If  the  length  and  breadth  are  measured  in  feet  and  A  in  square  feet, 
k  is  unity.  But,  if  L  and  B  are  measured  in  feet  and  A  in  acres, 
then  k  =  1/43560.  If  L  and  B  are  measured  in  rods  and  A  in 
acres,  then  k  =  1/160. 

From  Ohm's  law,  we  say  that  the  electric  current  in  a  circuit 
varies  directly  as  the  electromotive  force  and  inversely  as  the 
resistance,  or 

C  a  -51  or  C  =  A;  ^• 

K  K 

The  constant  multiplier  is  unity  if  C  be  measured  in  amperes,  E 
in  volts,  and  R  in  ohms,  so  that  for  these  units 

^  -  R 


Exercises 

1.  The  original  length  of  a  spring  is  10  inches.  The  force,  F, 
necessary  to  stretch  the  spring  is  directly  proportional  to  its  elongation, 
s.  (o)  Find  the  proportionality  factor  if  a  force  of  200  pounds  will 
hold  the  spring  at  a  length  of  12  inches,  (b)  What  force  will  be 
required  to  hold  the  spring  at  a  length  of  13  inches?     (c)  What  force 


§31]  THE  POWER  FUNCTION  69 

will  be  required  to  elongate  the  spring  1  inch?  Note  that  the  elon- 
gation is  the  extension  of  the  length  beyond  the  original  length  and  not 
the  total  length  after  elongation. 

StfGGBSTioN:  Since  the  force  F  is  directly  proportional  to  the  elon- 
gations, we  may  write 

F  =  ks, 

where  k  is  the  proportionality  factor.  We  have  given  that  F  is  200 
pounds  when  s  is  2  inches. 

2.  Hooke's  Law  states  that  the  elongation  of  a  steel  bar  is  propor- 
tional to  the  force  applied.  A  bar  500  inches  long  is  stretched  to  a 
length  of  500.5  inches  when  a  force  of  1000  pounds  is  applied.  Find 
the  proportionality  factor. 

3.  Boyle's  Law  states  that  the  volume  of  a  perfect  gas  at  constant 
temperature  varies  inversely  as  the  pressure.  If  volume  is  measured 
as  cubic  feet  and  pressure  as  pounds  per  square  inch,  find  the  pro- 
portionality factor  if  the  volume  is  13  cubic  feet  when  the  pressure 
is  60  pounds  per  square  inch.  What  will  be  the  volume  of  the  same 
gas,  according  to  Boyle's  Law,  if  the  pressure  becomes  only  15  pounds 
per  square  inch? 

31.  Illustrations  from  Science.  Some  of  the  most  important 
laws  of  natural  science  are  expressed  by  means  of  the  power  func- 
tion' or  graphically  by  means  of  loci  of  the  parabolic  or  hyperbolic 
type. 

The  linear  equation  y  =  mx  is,  of  course,  the  simplest  case  of  the 
power  function  and  its  graph,  the  straight  line,  may  be  regarded  as 
the  simplest  of  the  curves  of  the  parabolic  type.  The  following 
illustrations  will  make  clear  the  importance  of  the  power  function 
in  expressing  numerous  laws  of  natural  phenomena.  Later  the 
student  will  learn  of  two  additional  types  of  fundamental  laws  of 
science  expressible  by  two  functions  entirely  different  from  the 
power  function  now  being  discussed. 

The  instructor  will  ask  oral  questions  concerning  each  of  the 
following  illustrations.  The  student  should  have  in  mind  the 
general  form  of  the  graph  in  each  case,  but  should  remember  that 
the  law  of  variation,  or  the  law  of  change  of  value  which  the  func- 
tional relation  expresses,  is  the  matter  of  fundamental  importance. 
The  graph  is  useful  primarily  because  it  aids  to  form  a  mental  pic- 
ture of  the  law  of  variation  of  the  function.     The  practical  graph- 

1  For  brevity  ax"  as  well  as  a:"  will  frequently  be  called  a  power  function  of  x. 


70  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§31 

ing  of  the  concrete  illustrations  given  below  will  not  be  done  at 
present,  but  will  be  taken  up  later  in  §33. 

(a)  The  pressure  of  a  fluid  in  a  vessel  may  be  expressed  in  either 
pounds  per  square  inch  or  in  terms  of  the  height  of  a  column  of  mer- 
cury possessing  the  same  static  pressure.     Thus  we  may  write 

p  =  0.492/!,  (1) 

in  which  p  is  pressure  in  pounds  per  square  inch  and  h  is  the  height  of 
the  column  of  mercury  in  inches.  The  graph'  is  the  straight  line 
through  the  origin  of  slope  492/1000.  The  constant  0.492  can  be 
computed  from  the  data  that  the  weight  of  mercury  is  13.6  times 
that  of  an  equal  volume  of  water  and  that  1  cubic  foot  of  water 
weighs  62.5  pounds. 

In  this  and  the  following  equations,  it  must  be  remembered  that 
each  letter  represents  a  number,  and  that  no  equation  can  be  used  until 
all  the  magnitudes  involved  are  expressed  in  terms  of  the  particular 
units  which  are  specified  in  connection  with  that  equation. 

(6)  The  velocity  of  a  falling  body  which  has  fallen  from  a  state  of 
rest  during  the  time  t,  is  given  by 

V  =  32.2/,  (2) 

in  which  t  is  the  time  in  seconds  and  v  is  the  velocity  in  feet  per  second. 
If  t  is  measured  in  seconds  and  v  is  in  centimeters  per  second,  the 
equation  becomes'  v  =  98K.  In  either  case  the  graph  is  a  straight 
line,  but  the  lines  have  different  slopes. 

^  A  full  discuBsion  of  the  process  of  changing  formulas  like  the  ones  in  the  present 
section  into  a  new  set  of  units  should  be  sought  in  text-books  on  physics  and  mechan- 
ics. The  following  method  is  sufficient  for  elementary  purposes.  First,  write  (for 
the  present  example)  the  formula  v  =  32. 2£  where  v  is  in  ft./sec.  and  t  is  in  seconds. 
For  any  units  of  measure  that  may  be  used,  there  holds  a  general  relation  u  =  ct, 
where  c  is  a  constant.  To  determine  what  we  may  call  the  dimensions  of  c,  sub- 
stitute for  all  letters  in  the  formula  the  names  of  the  units  in  which  they  are  ex- 
pressed, treating  the  names  as  though  they  were  algebraic  numbers.  From  v  =  ct 
write,  ft./sec,  =  csec.  Hence  (solving  for  dimensions  of  c),  c  has  dimensions  ft./sec.^ 
Therefore  in  the  given  case,  we  know  c  =  32.2  ft. /sec. 2.  To  change  to  any  other 
units  simply  substitute  equals  for  equals.  Thus  1  ft.  =  30.5  cm.,  hence  c  =  32.2  X 
30.5  cm./sec.2  =  981  cm./sec.^ 

To  change  velocity  from  mi./hr.  to  ft./sec.  in  formula  (19)  below,  we  have  R  = 
0.00372  where  R  is  in  Ib./sq.  ft.  and  V  is  in  mi./hr.  Write  the  general  formula 
R  =  cY^.  The  dimensions  of  c  are  (lb./ft.2)  -7-  (mi.Vhr.2)  or  (lb./ft.2)  X  (hr.^/mi.sy. 
In  the  given  case  we  have  the  value  of  c  =  0.003  (lb. /ft. 2)  X  (hr.2/mi.2).  To  change 
V  to  ft./sec,  substitute  equals  for  equals,  namely  1  hr.  =  3600  sec,  1  mi.  =  5280 
ft.,  which  gives  the  formula  R  =  0.0013972^  where  V  is  expressed  as  ft./sec  and 
R  ig  expressed  as  lb./ft.2.     Note  that  1  mi./hr.  =  f  ft./sec.  approximately. 


§31]  THE  POWER  FUNCTION  71 

(c)  The  space  traversed  by  a  falling  body  is  given  by 

s  =  igt\  (3) 

or  in  English  itaits  (s  in  feet  and  t  in  seconds) 

s  =  16.1(2.       '  (4) 

(d)  The  velocity  of  the  f aUing  body,  from  the  height  h,  is 

V  =  ■\/2gh  =  V&iAh.  (5) 

The  resistance  of  the  air  is  not  taken  into  account  in  formulas  (2) 
to  (5). 

The  formula  equivalent  to  (5) 

jTOD^  =  mgh,  (6) 

where  to  is  the  mass  of  the  body,  expresses  the  equivalence  of  ^mv^, 
the  kinetic  energy  of  the  body,  and  mgh,  the  work  done  by  the  force  of 
gravity  mg,  working  through  the  distance  h. 

(e)  The  intensity  of  the  attraction  exerted  on  a  unit  mass  by  the 
sun  or  by  any  planet  varies  inversely  as  the  square  of  the  distance 
from  the  center  of  mass  of  the  attracting  body.  If  r  stand  for  that 
distance  and  if  /  be  the  force  exerted  on  unit  mass  of  the  attracted 
body,  then 

/  =  ^-  (7) 

The  constant  m  is  the  value  of  the  force  when  r  is  unity. 

(f)  The  formula  for  the  horse  power  transmissible  by  cold-roUed 
shafting  is 

where  H  is  the  horse  power  transmitted,  d  the  diameter  of  the  shaft  in 
inches,  and  N  the  number  of  revolutions  per  minute. 

The  rapid  increase  of  this  function  (as  the  cube  of  the  diameter) 
accounts  for  some  interesting  facts.  Thus  doubling  the  size  of  the 
shaft  operating  at  a  given  speed  increases  8-fold  the  amount  of  power 
that  can  be  transmitted,  while  the  weight  of  the  shaft  is  increased  but 
4-fold. 

If  H  be  constant,  N  varies  inversely  as  d^  Thus  an  old-fashioned 
5.0-h.p.  overshot  water-wheel  making  three  revolutions  per  minute 
requires  about  a  9-inch  shaft,  while  a  DeLaval  50-h.p.  steam  turbine 
making  16,000  revolutions  per  minute  requires  a  turbine  shaft  but 
little  over  J^  inch  in  diameter. 


72  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§31 

(g)  The  period  of  the  simple  pendulum  is 

T  =  irVrTg,  (9) 

where  T  is  the  time  of  one  swing  in  seconds,  I  the  length  of  the  pendu- 
lum in  feet,  and  g  =  32.2  ft. /sec.',  approximately. 

(h)  The  centripetal  force  on  a  particle  of  weight  W  pounds,  rotat- 
ing in  a  circle  of  radius  R  feet,  at  the  rate  of  JV  revolutions  per  second  is 

F  =  ^  ^"^  ,  (10) 

9 
or  if  ff  =  32.16  ft. /sec.?, 

F  =  1.227GWRN'  (11) 

where  F  is  measured  in  pounds.  If  N  be  the  number  of  revolutions 
per  minute,  then 

^  -      36009  ^^^^ 

=  0.000341  TFiJi\r2_  (13) 

(i)  An  approximation  formula  for  the  indicated  horse  power  required/ 
for  a  steamboat  is 

I.H.P.  =  ^,  (14) 

where  S  is  speed  in  knots,  D  is  displacement  in  tons,  and  C  is  a  con- 
stant appropriate  to  the  size  and  model  of  the  ship  to  which  it  is 
appUed.  The  constant  ranges  in  value  from  about  240,  for  finely 
shaped  boats,  to  200,  for  fairly  shaped  boats. 

(j)  Boyle's  law  for  the  expansion  of  a  gas  maintained  at  constant 
temperature  is 

pv  =  C,  (15) 

where  p  is  the  pressure  and  v  the  volume  of  the  gas,  and  C  is  a  constant. 
Since  the  density  of  a  gas  is  inversely  proportional  to  its  volume,  the 
above  equation  may  be  written  in  the  form 

P  =  cp,  (16) 

in  which  p  is  the  density  of  the  gas. 

(fc)  The  flow  of  water  over  a  trapezoidal  weir  is  given  by 

q=  Z.S7Lh^,  (17) 

where  q  is  the  quantity  in  cubic  feet  per  second,  L  is  the  length  of  the 
weir'  in  feet,  and  h  is  the  head  of  water  on  the  weir,  in  feet. 

I  The  instructor  is  expected  to  explain  the  meaning  of  the  terms  here  used. 


§31]  THE  POWEE  FUNCTION  73 

{I)  The  physical  law  holding  tor  the  adiabatic  expanBion  of  air, 
that  is,  the  law  of  expansion  holding  when  the  change  of  volume  is  not 
accompanied  by  a  gain  or  loss  of  heat/  is  expressed  by 

p  =  cp'-^''  (18) 

This  is  a  good  illustration  of  a  power  function  with  fractional  expo- 
nent. The  graph  is  not  greatly  different  from  the  semi-cubical 
parabola 

y  =  ci' 

(to)  The  pressure  or  resistance  of  the  air  upon  a  flat  surface  per- 
pendicular to  the  current  is  given  by  the  formula 

R  =  0.003F^  (19) 

in  which  V  is  the  velocity  of  the  air  in  miles  per  hour  and  B  is  the 
resulting  pressure  upon  the  surface  in  pounds  per  square  foot.  Ac- 
cording to  this  law,  a  20-mile  wind  would  cause  a  pressure  of  about  1.2 
pounds  per  square  foot  upon  the  flat  surface  of.  a  building.  One  foot 
per  second  is  equivalent  to  about  2/3  mile  per  hour,  so  that  the  formula 
when  the  velocity  is  given  in  feet  per  second  becomes : 

R  =  0.0013F2.  (20) 

(n)  The  power  used  to  drive  an  aeroplane  may  be,  divided  into  two 
portions.  One  portion  is  utilized  in  overcoming  the  resistance  of  the 
air  to  the  onward  motion.  The  other  part  is  used  to  sustain  the 
aeroplane  against  the  force  of  gravity.  The  first  portion  does  "use- 
less" work — ^work  that  should  be  made  as  small  as  possible  by  the 
shapes  and  sizes  of  the  various  parts  of  the  machine.  The  second  part 
of  the  power  is  used  to  form  continuously  anew  the  wave  of  compressed 
air  upon  which  the  aeroplane  rides.  Calling  the  total  power''  P,  the 
power  required  to  overcome  the  resistance  Pr,  and  that  used  to  sus- 
tain the  aeroplane  P«,  we  have 

P  =Pr+P,  (21) 

We  learn  from  the  theory  of  the  aeroplane  that  P,  varies  as  the  cube 
of  the  velocity,  while  P,  varies  inversely  as  V,  so  that 

Pr  =  cV^  (22) 

^  Note  that  when  a  vessel  containing  a  gas  is  insulated  by  a  non-conductor  of 
heat,  so  that  no  heat  can  enter  or  escape  from  the  vessel,  that  the  temperature  of  the 
gaa  will  rise  when  the  gas  is  compressed,  and  fall  when  it  is  expanded.  Adiabatic 
expansion  may  be  thought  of,  therefore,  as  taking  place  in  an  inaulated  vessel. 

2  Power  (=  work  done  per  unit  time)  is  measured  by  the  unit  horse  power,  which 
ia  550  foot-pounds  per  second. 


74  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§31 


and 


P,  = 


k 


(23) 


Thus  at  high  velocity  less  and  less  power  is  requireii  to  sustain  the 
aeroplane  but  more  and  more  is  required  to  overcome  the  frictional 
resistance  of  the  medium.  The  law  expressed  by  (23)  that  less  and 
less  power  is  required  to  sustain  the  aeroplane  as  the  speed  is  increased 
is  known  as  Langley's  Law.     From  this  law  Langley  was  convinced 

25 
24 
.23 

S22 
S21 
S20 

§19 
h18 

|17 
gl6 

3  15 
014 

1 13 
^112 
Sll 

Sio 

39 

g  8 

«  6 
■g  5- 

fl  4 

a 
Hi  2 

1 


in  1  T  iL_r  rr  r  /^ 

M-   t  '^t   t  T _/J^z 

3L  txi  t^^ 4  J  Z^^/ 

W^Tf/   '!   /b/ /-7/   / 

^l^LJ    ti    Ij^Xj^ ^^ 

4ZJ-t   tJ  L/t/y^V 

4  IJjty  r  /.ryY/^ 

J    ^1^4    Tl_/^Tl/Zy& 

ti^  tt-i/tj/L^ui^TO 

ttl--4J^'tij.^\ftiy^4^ 

ttttj^-/^tKttZZC^ 

'^trr'^ttZt^Z-^t^ 

^  XlTH-JltZC^"^^^^ 

H  37^^555277^^53?^ 

Iir77Z55/:522^;^2^ 

tinmlt2z4>^t^ 

[I/QZ6Z232|g?^ 

IDAZgggglpJ^ 

ffizzpppp^ 

Wl  /Aw^,  ^^' 

m™  ^^^^^ 

J      aj^w 

L 

t 

)Tl*lO«DC-000>o,-IOaCQ-3'»rt(ot-d60»^ 


Gals.for  One  Foot  Depth 

Fig.  42. — Capacity  of   rectangular  and    circular  tanks  per  foot  of 

depth. 


that  artificial  flight  was  possible,  for  the  whole  matter  seemed  to 
depend  primarily  upon  getting  up  sufficient  speed.  It  is  really  this 
law  that  makes  the  aeroplane  possible.  An  analogous  case  is  the 
well-known  fact  that  the  faster  a  person  skates,  the  thinner  the  ice 
necessary  to  sustain  the  skater.  In  this  case  part  of  the  energy  of 
the  skater  is  continually  forming  anew  on  the  thin  ice  the  wave  of 
depression  which  sustains  the  skater,  while  the  other  part  overcomes 
the  frictional  resistance  of  the  skates  on  the  ice  and  the  resistance  of 
the  air. 


§32]  THE  POWER  FUNCTION  75 

(o)  The  capacity  of  cast-iron  pipe  to  transmit  water  is  often  given 
by  the  formula 

9'-88  =  1.68W-"  (24) 

in  which  q  is  the  quantity  of  water  discharged  in  cubic  feet  per  second, 
d  is  the  diameter  of  the  pipe  in  feet,  and  h  is  the  loss  of  head  measured 
in  feet  of  water  per  1000  linear  feet  of  pipe.  This  is  a  good  illustra- 
tion of  the  equation  of  a  parabolic  curve  with  complicated  fractional 
exponents.  The  curve  very  roughly  approximates  the  locus  of  the 
equation 

y  =  cVhxi.  (25) 

(p)  The  contents  in  gallons  of  a  rectangular  tank  per  foot  of  depth, 
6  feet  wide  and  I  feet  long,  is 

q  =  7.5W.  (26) 

The  contents  in  gallons  per  foot  of  depth  of  "a  cylindrical  tank  d  feet  in 
diameter  is 

q  =  7.5^^74.  (27) 

Fig.  42  shows  the  graph  of  (26)  for  various  values  of  6  and  also  shows 
to  the  same  scale  the  graph  of  (27). 

32.  Rational  and  Empirical  Equations. — A  number  of  the 
formulas  given  above  are  capable  of  demonstration  by  means  of 
theoretical  considerations  only.  Such  for  example  are  equations 
(1),  (2),  (3),  (4),  (5),  (7),  (8),  (9),  (10),  etc.,  although  the  constant 
coefficients  in  many  of  these  cases  were  experimentally  deter- 
mined. Formulas  of  this  kind  are  known  in  mathematics  as 
rational  equations.  On  the  other  hand  certain  of  the  above  for- 
mulas, especially  equations  (14),  (17),  (19),  (22),  (23),  (24), 
including  not  only  the  constant  coefficients  but  also  the  law  of 
variation  of  the  function  itself,  are  known  to  be  true  only  as  the 
result  of  experiment.  Such  equations  are  called  empirical 
equations.  Such  formulas  arise  in  the  attempt  to  express  by  an 
equation  the  results  of  a  series  of  laboratory  measurements. 

For  example,  the  density  of  water  (that  is,  the  mass  per  cubic 
centimeter  or  the  weight  per  cubic  foot)  varies  with  the  tem- 
perature of  the  water.  A  large  number  of  experimentors  have 
prepared  accurate  tables  of  the  density  of  water  for  wide  ranges 
of  temperature  centigrade,  and  a  number  of  very  accurate  empir- 
ical formulas  have  been  ingeniously  devised  to  express  the  results, 
of  which  the  following  four  equations  are  samples : 


76  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§33 

Empirical  fonnulas  jor  the  density,  d,  of  water  in  terms  of  tem- 
perature centigrade,   B. 

96(9  -  4)2 


(a)    d  =  1  - 


10' 


(K^    ^       1        93(0  -  4)i»«^ 
(6)     d  =  1 jq^^ 

,  ,     J       ,       6fl2  -  369  +  47 
(c)    "^  =  1 io« 

,  „     J        ,     ,   0.4859»  -  81.39^  +  6029  -  1118 
(d;    d  =  1  H jq^ 

Exercises 

1.  Among  the  power  functions  named  in  the  above  illustrations, 
pick  out  examples  of  increasing  functions  and  of  decreasing  functions. 

2.  Under  the  same  difference  of  head  or  pressure,  show  by  formula 
(24)  that  an  8-inch  pipe  will  transmit  much  more  than  double  the 
quantity  of  water  per  second  that  can  be  transmitted  by  a  4-inch  pipe. 

3.  Wind  velocities  during  exceptionally  heavy  hurricanes  on  the 
Atlantic  coast  are  sometimes  over  140  miles  per  hour.  Show  that  the 
wind  pressure  on  a  flat  surface  during  such  a  storm  is  about  fifty  times 
the  amount  experienced  during  a  20-mile  wind. 

4.  Show  that  for  wind  velocities  of  10,  20,  40,  80,  160  miles  per  hour 
(varying  in  geometrical  progression  with  ratio  2),  the  pressure  exerted 
on  a  flat  surface  is  0.3,  1.2,  4.8,  19.2,  76.8  pounds  per  square  foot 
respectively  (varying  in  geometrical  progression  with  ratio  4) . 

6.  A  300-h.p.  DeLaval  turbine  makes  10,000  revolutions  per  min- 
ute.    Find  the  necessary  diameter  of  the  propeller  shaft. 

6.  A  railroad  switch  target  bent  over  by  the  wind  during  a  tornado 
in  Minnesota  indicated  an  air  pressure  due  to  a  wind  of  600  miles  per 
hour.  Show  that  the  equivalent  pressure  on  a  flat  surface  would  be 
7.5  pounds  per  square  inch. 

7.  Show  that  a  parachute  50  feet  in  diameter  and  weighing  50 
pounds  will  sustain  a  man  weighing  205  pounds  when  falling  at  the 
rate  of  10  feet  per  second. 

Suggestion:  Use  approximate  value  ir  =  22/7  in  finding  area  of 
parachute  from  formula  for  circle,  nr^,  and  use  formula  (20)  above. 

8.  Show  that  empirical  formulas  (a)  and  (6)  for  the  density  of 
water  reduce  to  a  power  function  if  the  origin  be  taken  at  9  =  4,  d  =  1. 

33.  Practical  Graphs  of  Power  Functions.  The  graphs  of  the 
power  function 

2/  =  a;^    y  =  x^,    y  =  ->    y  =  x\,    etc.,  (1) 


§33] 


THE  POWER  FUNCTION 


77 


can,  of  course,  be  made  the  basis  of  the  laws  concretely  expressed 
by  equations  (1)  to  (27)  of  §31.  If,  however,  the  graph  of  a 
scientific  formula  is  to  serve  as  a  numerical  table  of  the  function 
for  actual  use  in  practical  work,  then  there  is  much  more  labor 
in  the  proper  construction  of  the  graph  than  the  mere  plotting 
of  the  abstract  mathematical  function.  The  size  of  the  unit  to 
be  selected,  the  range  over  which  the  graph  should  extend,  the 
permissible  course  of  the  curve,  become  matters  of  practical 
importance. 


If  the  apparent  slope^  of 
+  1  or  —1,  it  is  desirable 
to  make  an  abrupt  change 
of  unit  in  the  vertical  or 
the  horizontal  scale,  so  as 
to  bring  the  curve  back 
to  a  desirable  course,  for 
it  is  obvious  that  numeri- 
cal readings  can  best  be 
taken  from  a  curve  when 
it  crosses  the  rulings  of  the 
coordinate  paper  at  ap- 
parent slopes  differing  but 
little  from  + 1. 

The   above   suggestions 
in  practical  graphing  are 
the  follow- 


a  graph  departs  too  widely  from 


E 
350 


■ 

i 

' 

1 

\ 

/ 

' 

/ 

/ 

( 

' 

> 

1 

/ 

' 

> 

/ 

/ 

/ 

ya 

-A 

7 

ir 

J 

2       3       4       5      6      7       8 
Diameter  of  Shaft  in  Inches 


9     10 


Pig.  43.— Capacitjr  at  100  R.P.M. 
of  cold-rolled  shafting  to  transmit 
power. 


illustrated  by 
ing  example : 

Graph       the      formula 
(equation  (8),  §31),  for  the  horse  power  transmissible  by  cold- 
rolled  shafting. 


in  which  d  is  the  diameter  in  inches  and  N  is  the  number  of 
revolutions  per  minute.  The  formula  is  of  interest  only  for  the 
range  of  d  between  0  and  24  inches,  as  the  dimensions  of  ordinary 

1  Of  course  the  real  slope  of  a  curve  is  independent  of  the  scales  used.  By  apparent 
slope  =3  1  is  meant  that  the  graph  appears  to  cut  the  ruling  of  the  squared  paper 
at  about  45°. 


78  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§33 

shafting  lie  well  within  these  limits.  Likewise  one  would  not 
ordinarily  be  interested  in  values  of  N  except  those  lying  between 
10  and  3000  revolutions  per  minute.  Fig.  43  shows  a  suitable 
graph  of  this  formula  for  the  range  1  <  d  <  10  for  the  fixed 
value  oiN  =  100.  In  order  properly  to  graph  this  function,  three 
different  scales  have  been  used  for  the  ordinate  H,  so  that  the 
slope  of  the  curve  may  not  depart  too  widely  from  unity. 

If  similar  graphs  be  drawn  for  N  =  200,  N  =  300,  N  =  400, 
etc.,  a  set  of  parabolas  is  obtained  from  which  the  horse  power 
of  shafting  for  various  speeds  of  rotation  as  well  as  for  various 
diameters  may  be  obtained  at  once.  A  set  of  curves  systematic- 
ally constructed  in  a  manner  similar  to  that  just  described,  is  often 
called  a  family  of  curves.  Fig.  42  shows  a  family  of  straight  lines 
expressing  the  capacity  of  rectangular  tanks  corresponding  to 
the  various  widths  of  the  tanks. 

Inasmuch  as  many  of  the  fqrmulas  of  science  are  used  only  for 
positive  values  of  the  variables,  it  is  only  necessary  in  these  cases 
to  graph  the  function  in  the  first  quadrant.  For  such  problems 
the  origin  may  be  taken  at  the  lower  left  corner  of  the  coordinate 
paper  so  that  the  entire  sheet  becomes  available  for  the  curve  in 
the  first  quadrant. 

The  illustrations  of  §31  are  sufficient  to  make  clear  the  impor- 
tance in  science  of  the  functions  now  being  discussed.  The  follow- 
ing exercises  give  further  practice  in  the  useful  application  of  the 
properties  of  the  functions. 

Exercises 

The  graphs  for  the  following  problems  are  to  be  constructed  upon 
rectangular  coordinate  paper.  The  instructions  are  for  centimeter 
paper  (form  Ml)  ruled  into  20  X  25  cm.  squares.  On  other  paper  use 
J  inch  in  place  of  1  centimeter.  In  each  case  the  units  for  abscissa 
and  for  ordinates  are  to  be  so  selected  as  best  to  exhibit  the  functions, 
considering  both  the  workable  range  of  values  of  the  variables  and  ' 
the  suitable  slope  of  the  curves. 

The  student  should  read  §12  a  second  time  before  proceeding  with 
the  following  exercises,  giving  especial  care  to  instructions  (4),  (5),  and 
(6)  of  that  section. 

1.  Classify  the  graphs  of  formulas  (1)  to  (27),  §31,  as  to  parabolic 
or  hyperbolic  type. 


§33]  THE  POWER  FUNCTION  79 

2.  Graph  the  formula  v^  =  2gh,  or  v  =  y/lgh  =  8.02hi,  if  h  range 
between.  1  and  100,  the  second  and  foot  being  the  units  of  measure. 
See  formula  (5),  §31. 

The  following  table  of  values  is  readily  obtained : 

h\  1  5  10  20  30  40  50  60  70  80  90  100 
v\   8.02    17.9   25.3   35.8   43.9   50.7   56.7   62.1   67.1   71.7   76.0   80.2 

Use  2  cm.  =  10  feet  as  the  horizontal  unit  for  h,  and  2  cm.  =  10  feet 
per  second  as  the  vertipal  unit  for  v.  The  graph  is  then  readily  con- 
structed without  change  of  unit  or  other  special  expedient.^ 

3.  Graph  the  formula  q  =  3.37Lhi  fori  =  1,  and  for  h  =  0,  0.1, 
0.2,  0.3,  0.4,  0.5.  See  formula  (17),  §31.  Use  4  cm.  =  0.1  for 
horizontal  unit  for  h  and  2  cm.  =0.1  for  vertical  unit  for  q. 

4.  Draw  a  curve  showing  the  indicated  horse  power  of  a  ship  I.H.P. 
=  S'Di/C  for  C  =  200  if  the  displacement  D  =  8000  tons,  and  for 

the  range  of  speeds  iS  =  10  to  S  =  20  knots.     See  formula  (14),  §31. 
For  the  vertical  unit  use  1  cm.  —  1000  h.p.  and  for  the  horizontal 
unit  use  2  cm.  =  1  knot.     Call  the  lower  left-hand  corner  of  the  paper 
the  point  (S  =  10,  I.H.P.  =  0). 

5.  From  the  formula  expressing  the  centripetal  force  in  pounds  of  a 
rotating    body, 

F  =  0.000341  ITiJiV^ 

draw  a  curve  showing  the  total  centripetal  force  sustained  by  a  36-inch 
automobile  tire  weighing  25  pounds,  for  all  speeds  from  10  to  40  miles 
per  hour.     See  formula  (13),  §31. 

Miles  per  hour  must  first  be  converted  into  revolutions  per  minute 
by  .dividing  5280  by  the  circumference  of  the  tire  and  then  dividing 
the  result  by  60.     This  gives 

1  mile  per  hour  =  9J  revolutions  per  minute 

If  V  be  the  speed  in  miles  per  hour  the  formula  for  F  becomes 

F  =  0.000341(1.5)25(9J)2F2  =  l.llF^ 

For  horizontal  scale  let  4  cm.  =  10  miles  per  hour  and  for  the  vertical 
scale  let  1  cm:  =  100  pounds. 

6.  Draw  a  curve  from  the  formula  /  =  m/r'^  showing  the  accelera- 
tion of  gravity  due  to  the  earth  at  all  points  between  the  surface  of 
the  earth  and  a  point  240,000  miles  (the  distance  to  the  moon)  from 
the  center,  if  /  =  32.2  when  radius  of  the  earth  =  4000  miles. 

It  is  convenient  in  constructing  this  graph  to  take  the  radius  of 
the  earth  as  unity,  so  that  the  graph  will  then  bo  required  of  /  =  32.2/r^ 


80 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§34 


from  r  ■«  1  to  r  =  60.  In  order  to  construct  a  suitable  curve  several 
changes  of  units  are  desirable.  See  Kg.  44.  One  centimeter  repre- 
sents one  radius  (4000  miles)  from  r  =  0  to  r  =  10,  after  which  the 
scale  is  reduced  so  that  1  mm.  represents  one  radius.  In  the  vertical 
direction  the  scale  is  4  cm.  =  10  feet  per  second  for  0  <  r  <  5,  4  cm.  = 
1  foot  a  second  for  5  <  r  <  10,  and  4  cm.  =0.1  foot  a  second  for 
10  <  r  <  60.  Even  with  these  four  changes  of  units  just  used  the 
first  and  third  curves  are  somewhat  steep.  The  student  can  readily 
improve  on  the  scheme  of  Fig.  44  by  a  better  selection  of  units. 


40 


80 


20 


90 

I 

I 

\ 

1 

I 

\ 

N> 

\ 

\ 

S 

' 

V 

^ 

\, 

\ 

^, 

--. 

\ 

s 

"^ 

^ 

— 

g  43 

o 

I      10 


1    2    3  4    6    6    7    8    9  10  20  SO  40  EO  60 
Distance  from  Earth's.  Oeuter,  Earth's  Eadiuscl 

Fig.  44. — Gravitational  acceleration  at  various  distances  from  the 
earth's  center.  The  moon  is  distant  approximately  60  earth's  radii 
from  the  center  of  the  earth. 


34.  The  Law  of  the  Power  Functions.  Sufficient  illustrations 
have  been  given  to  show  the  fundamental  character  of  the  power 
function  as  an  expression  of  numerous  laws  of  natural  phenomena. 
How  may  a  functional  dependence  of  this  sort  be  expressed  in 
words?  If  a  series  of  measurements  are  made  in  the  laboratory, 
so  as  to  produce  a  numerical  table  of  data  covering  certain  phe- 
nomena, how  can  it  be  determined  whether  or  not  a  power  func- 
tion can  be  written  down  which  will  express.the  law  (that  is,  the 
function)  defined  by  the  numerical  table  of  laboratory  results? 


§34]  THE  POWER  FUNCTION  81 

The  answers  to  these  questions  are  readily  given.    Consider  first 
the  law  of  the  falling  body 

s  =  16.1i^  (1) 

Make  a  table  of  values  for  values  oi  t  =  1,  2,  4,  8,  16  seconds,  as 
follows : 


t 

1 

2 

4 

8 

16 

s 

16.1 

64.4 

257.6 

1030.4 

4121.6 

The  values  of  t  have  been  so  selected  that  t  increases  by  a  fixed 
multiple,  that  is,  each  value  of  t  in  the  sequence  is  twice  the  pre- 
ceding value.  From  the  corresponding  values  of  s  it  is  observed 
that  s  also  increases  by  a  fixed  multiple,  namely  4. 

Similar  conclusions  obviously  hold  for  any  power  function. 
Take  the  general  case 

y  =  ax",  (2) 

where  n  is  any  exponent,  positive,  negative,  integral  or  fractional. 
Let  X  change  from  any  value  xi  to  a  multiple  value  mxi  and  call 
the  corresponding  values  of  y,  yi  and  2/2.    Then  we  have 

2/1  =  axi",  (3) 

and 

2/2  =  o(wia;i)"  =  aiwxi".  (4) 

Divide  the  members  of  (4)  by  the  members  of  (3)  and  we  have 

^  =  m».  (5) 

2/1 

That  is,  if  a;  in  any  power  function  change  by  the  fixed  multiple 
m,  then  the  value  of  y  will  change  by  a  fixed  multiple  w.  Thus 
the  law  of  the  power  function  may  be  stated  in  words  in  either  of 
the  two  following  forms : 

In  any  power  function  of  x,  if  x  change  by  a  fixed  multiple,  y  will 
change  by  a  fixed  multiple  also. 

In  any  power  function  of  x,  if  x  increase  by  a  fixed  percent, 
the  function  will  increase  or  decrease  by  a  fixed  percent  also. 

This  test  may  readily  be  applied  to  laboratory  data  to  determine 
whether  or  not  a  power,  function  can  be  set  up  to  represent  as  a 
formula  the  data  in  hand.  To  apply  this  test,  select  at  several 
places  in  one  column  of  the  laboratory  data,  pairs  of  numbers 
which  change  by  a  selected  fixed  percent,  say  10  per  cent,  or  20 


82  ELEMENTARY  MATHEMATICAL  ANALYSIS        1§35 

percent,  or  any  convenient  percent.  Then  the  corresponding  pairs 
of  numbers  in  the  other  column  of  the  table  must  also  be  related  by 
a  fixed  percent  (of  course,  not  in  general  the  same  as  the  first- 
named  percent),  provided  the  functional  relation  is  expressible  by 
means  of  a  power  function.  If  this  test  does  not  succeed,  then 
the  function  in  hand  is  not  a  power  function. 

Since  the  fixed  percent  for  the  function  is  ot"  if  the  fixed  percent 
for  the  variable  be  m,  the  possibility  of  determining  n  exists, 
since  the  table  of  laboratory  data  must  yield  the  numerical  values 
of  both  m  and  to" 

36.  Simple  Modifications  of  the  Parabolic  and  of  the  H]rperbolic 
Types  of  Curves.  In  the  study  of  the  motion  of  objects  it  is 
convenient  to  divide  bodies  into  two  classes:  first,  bodies  which 
retain  their  size  and  shape  unaltered  during  the  motion;  second, 
bodies  which  suffer  change  of  size  or  shape  or  both  during  the 
motion.  The  first  class  of  bodies  are  called  rigid  bodies ;  a  mov- 
ing stone,  the  reciprocating  or  rotating  parts  of  a  machine,  are 
illustrations.  The  second  class  of  bodies  are  called  elastic  bodies ; 
a  piece  of  rubber  during  stretching,  a  spring  during  elongation  or 
contraction,  a  rope  or  wire  while  being  coiled,  the  water  flowing  in 
a  set  of  pipes,  are  all  illustrations  of  this  class  of  bodies. 

When  a  body  changes  size  or  shape  the  motion  is  called  a 
strain. 

Bodies  that  preserve  their  size  and  shape  unchanged  may  possess 
motion  of  two  simple  types:  (1)  Rotation,  in  which  all  particles 
of  the  body  move  in  circles  whose  centers  lie  in  a  straight  fine 
called  the  axis  of  rotation,  which  line  is  perpendicular  to  the  plane 
of  the  circles,  and  (2)  translation,  in  which  every  straight  line  of 
the  body  remains  fixed  in  direction. 

We  have  already  noted  that  the  curve. 

(1) 


2/1  = 

ax' 

or 

a 

s". 

can 

be  made  from  the 

curve 

y  = 

x" 

(2) 
by  multiplying  all  the  ordinates  of  (2)  by  a.     The  effect  is  either 


§36]  THE  POWER  FUNCTION  83 

to  elongate  or  to  contract  all  of  the  ordinates,  depending  upon 
whether  a  >  1  or  a  <  1,  respectively.     The  substitution  of  (j/i/a) 
for  y  has  therefore  produced  a  motion  or  strain  in  the  curve 
y  =  x".    Likewise 

»=(?)■  (« 

can  be  made  from 

2/  =  X"  (4) 

by  multiplying  all  of  the  abscissas  of  (4)  by  a.     The  effect  is 
either  to  stretch  or  to  contract  all  of  the  abscissas,  depending 
upon  whether  a  >  1,  or  a  <  1,  respectively. 
In  general,  if  a  curve  has  the  equation 

V  =  fix),  (5) 

then 

(6) 


»=/(?) 


is  nlade  from  curve  (5)  by '  lengthening  or  stretching  the  XY- 
plane  uniformly  in  the  x  direction  in  the  ratio  1  :  a. 

The  statement  just  given  is  made  on-  the  assumption  that 
a>l.  If  a<l  then  the  above  statements  must  be  changed 
by  substituting  shorten  or  contract  for  elongate  or  stretch. 

The  reasons  for  the  above  conclusions  have  been  previously 

stated :  substituting  (— )  everywhere  in  the  place  of  x  multiplies 
all  of  the  abscissas  by  a.     That  is,  if  ( — I  =  x,  then  xi  =  ax,  so  that 

Xi  is  a-fold  the  old  x. 

We  shall  now  explain  how  certain  of  the  other  motions  men- 
tioned above  may  be  given  to  a  locus  by  suitable  substitution  for 
X  and  y. 

36.  Translation  of  Any  Locus.     If  a  table  of  values  be  prepared 


for  each  of  the  equations 
as  follows : 


y  =  x'  (1) 

2/  =  (xi  -  3)2  (2) 


x  1 

-2     -10     12     3    4] 

y  1 

1 

4         1     0     1     4    9  16 
-2     -10     12    3     4     5     6 

Xi 

2/1       25       16     9    4     1     0     1     4     9 


84  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§36 

and  then  if  the  graph  of  each  be  drawn,  it  will  be  seen  that  the 
curves  differ  ordy  in  their  location  and  not  at  all  in  shape  or  size. 
The  reason  for  this  is  obvious:  If  {xi  —  3)  be  substituted  for  x 
in  any  equation,  then  since  {xi  —  3)  has  been  put  equal  to  x,  it 
follows  that  a;i  =  a;  +  3,  or  the  new  x,  namely  Xi,  is  greater 


Fig.  45. — The  curve  y^  =  (x  —  f)^  is  the  curve  y'  =  x'  translated 
to  the  right  :|  units. 


than  the  original  x  by  the  amount  3.  This  means  that  the  new 
longitude  of  each  point  of  the  locus  after  the  substitution  is  greater 
than  the  old  longitude  by  the  "fixed  amount  3.  Therefore  the 
new  locus  is  the  same  as  the  original  locus  translated  to  the  right 
the  distance  3. 


§36]  THE  POWER  FUNCTION  85 

The  same  reasoning  applies  if  {xi  —  a)  be  substituted  for  x, 
and  the  amount  of  translation  in  this  case  is  a.  The  same  reason- 
ing applies  also  to  the  general  case  y  =  f{x)  and  y  =  f{xi  —  a), 
the  latter  curve  being  the  same  as  the  former,  translated  the  dis- 
tance a  in  the  x  direction. 

As  it  is  always  easy  to  distinguish  from  the  context  the  new  x 
from  the  old  x,  it  is  not  necessary  to  use  the  symbol  Xi,  since  the 
old  and  new  abscissas  may  both  be  represented  by  x.  The 
following  theorems  may  then  be  stated: 

Theorems  on  Loci 

XI.  If  {x  —  a)  be  substituted  for  x  throughout  any  equation,  the 
ecus  is  translated  a  distance  a  in  the  x  direction. 

XII.  If  {y  —  6)  be  substituted  for  y  in  any  equation,  the  locus  is 
translated  the  distance  b  in  the  y  direction. 

These  statements  are  perfectly  general:  if  the  signs  of  a  and 
6  are  negative,  so  that  the  substitutions  for  x  and  y  are  of  the  form 
X  +  a'  and  y  +  b',  respectively,  then  the  translations  are  to 
the  left  and  down  instead  of  to  the  right  and  up. 

Sometimes  the  motion  of  translation  may  seem  to  be  disguised 
by  the  position  of  the  terms  a  or  b.  Thus  the  locus  y  =  3x  -\-  5 
is  the  same  as  the  locus  y  ~  Sx  translated  upward  the  distance  5, 
for  the  first  equation  is  really  y  —  5  =  3x,  from  which  the  conclu- 
sion is  obvious. 

Exercises 

The  student  is  not  required  to  draw  the  curves  in  exercises  1  to  8  below, 
but  is  expected  to  make  the  comparisons  by  means  of  the  theorems  on  loci 
given  above. 

1.  Compare  the  curves:  (1)  y-  -  2x  and  y  =  2(z  —  I);  (2)  y  =  x' 
and  2/  =  (x  —  4)';  (3)  y  =  x^  and  y  —  Z  =  x^;  (4)  ?/  =  x^  and 
y  =  (x  -  5)1;  (5)  y  =  5x^  and  y  =  5{x  +  3)2;  (6)  y  =  2x'  and 
y  =  2{x  -  fc)';  (7)  y  =  2x'  and  y  =  2x'  +  k;  (8)  y  +  7  =  x'  and 
y  =  x'  and  y  -  7  =  x';  (9)  Sy"  =  Sx'  and  3(,y  -  6)^  =  5(x  -  a)\ 

2.  Compare  the  curves:  (1)  y  =  x^  and  y  =  {x/2)^;  (2)  y  =  x^ 
and  y  =  x^/8;  (3)  y  —  x'  and  y/2  =  x';  (4)  y  =  x^  and  y  =  2x';  (5) 
2/2  =  3x«  and  {.y/bY  =  3(x/7)3;  (6)  y^  =  x'  and  y'^  =  (3i)';  (7) 
2/  =  x*  and  y  —  ix^  (note:  explain  in  two  ways);  (8)  y  =  x^  and 
2y  =  x'  and  y  =  27x». 


86  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§36 

3.  Translate  the  locus  y  =  2a:';  (1)  3  units  to  the  right;  (2)  4  units 
down;  (3)  5  units  to  the  left. 

4.  Elongate  three-fold  in  the  x  direction  the  loci:  (1)  y^  =  x;  (2) 
3y  =  x^;  (3)  y^  =  2x>;  (4)  y  =2x  +  7. 

6.  The  loci  named  in  exercise  4  have  their  ordinates  shortened  in 
the  ratio  2:1;  write  their  equations. 

6.  Show  that  y  =  — -r-r  and  y  = r  are  hyperbolas. 

X  ~i~  0  X         o 

7.  Show  that  y  =  — -j-r^  is  a  hyperbola. 

Note  :    Divide  the  numerator  by  the  denominator,  obtaining  the 

b  b 

equation  y  =  1 ^—r>  oi  y  —  1  =  — 


8.  Show  that    y  =^ 


or 


x+b'"'"       '  -       x+b 
X  +  a 
xH>' 

a  —  b 


y  =  i+: 


x+b 

is  a  hyperbola,   namely  the  curve  xy  =  a  —  b  translated  to  a  new 
position. 

«    oi    i  1        ,  N  ^         ,i_s  ^  +  3     ,  ^  3x  +  2  , 

9.  Sketch:     (a)  y  =  ^-^7^;   (b)  y  =  ^qrj;   (c)    V  =  "Jipj";   ^"" 

id)  y  =  — 3-0"'     Sketch  a  curve  from  which  each  curve  is  obtained 

by  translation. 

10.  Show  how  the  graph  for  t/  =  x^  +  4:X  +  5  may  be  obtained  from 
the  graph  for  y  =  x". 

Hint:  2/  =  x^  +  4x  +  5  =  x^  +  4x  +  4  +  1  =  (x  +  2)^  +  1,  or 
2/  —  1  =  (x  +  2)2.  Thus,  the  graph  for  y  =  x'  +  4:X  +  5  may  be 
obtained  by  translating  the  graph  for  y  =  x^  one  unit  up  and  two  units 
to  the  left. 

11.  Sketch  the  curves  for: 

(a)  2/  =  x2  +  4x  +  4;  (b)  y  =  x^  +  6x  +  10; 

(c)  2/  =  x2  +  2x  -  3;  (d)  2/  =  4x2  4.  4^;  +  j. 

(e)   y  =  4x'  +  2x  -  1;  (f)  y  =  ^x  -  x"; 

(s)  2/  =  6x  -  x";  (h)  2/  =  x^  +  3x  -  1; 

(i)   y  ^2x^  +Zx;  (j)  j/  =  3x  -  2x2; 
(fc)  2/^  =  X  +  1. 

12.  Which  of  the  curves  of  exercise  11  pass  through  the  origin? 

13.  Sketch: 

(a)  x2  +2/2  =  1;  (6)  x^  +  j/^  =  4; 

(c)  x2  +  (2/  -  i;2  =  1;  (d)  (x  -  1)2  +  2/2  =  4; 

(e)    (x  +  1)2  +  (y  -  2)2  =  5;  (/)   x2  +  2x  +  2/*  =  3. 


§37] 


THE  POWER  FUNCTION 


87 


37.  Shearing  Motion.  Let  the  dotted  curve  Pi'OPi,  Fig.  46 
be  the  graph  of  the  semicubical  parabola  y  =  xi  and  OP2  the  graph 
of  y  "=  X.  The  graph  P'OP  is  constructed  by  taking  its  ordinate, 
for  any  value  of  the  abscissa, 
equal  to  the  (algebraic)  sum  of 
the  ordinates  of  the  two  given 
curves.  Thus,  DP  =  DPi  + 
DPi  and  DP'  =  DPi'  +  DP,. 
The  equation  of  P'OP  is  y  = 
a;t  +  X,    since    DPi  =  xi    and 

DP2  =  X. 

Exercises 

1.  From  the  curves  for  y  =  x^ 
and  y  =  ^x,  sketch  y  =  x^  -\-  |.-c. 

2.  From  the  curves  for  y  =  x'^ 
and  2/  =  —  |x,  sketch  y  =  x''  — 
^x. 

3.  From  the  curves  for  y  =  — 
x^  and  y  =  x,  sketch  y  =  x  —  x^. 

4.  From  the  curves  for  y  =  - 

X 

and  y  =  X,  sketch  y  =  — \-  x- 

5.  From  the  curves  for  y  =  - 

X  ■ 

and  y  =  X,  sketch  y  =  x 

38.  General  Case.  Consider 
the  production  of  the  curve 

y  =  fix)  +  mx  (1)' 

from  the  curve 


Fig 


— The  shear  of  y 
the  line  y  =  x. 


and  the  straight  line 


y'  =  m 


(2) 

(3) 

Graphically,  the  curve  (1)  is  seen  to  be  formed  by  the  addition 
of  the  ordinates  of  the  straight  line  y"  =  mx  to  the  corresponding 
ordinates  of  y'  =  f(x) .  Thus,  in  Fig.  47,  the  graph  of  the  func- 
tion a;^  +  a;  is  made  by  adding  the  corresponding  ordinates  of 


88 


ELEMENTARY  MATHEMATICAL  ANALYSIS       [§38 


y  =  x^  and  y  =  x.  Mechanically,  this  might  be  done  by  draw- 
ing the  curve  on  the  edge  of  a  pack  of  cards  (see  Fig.  48),  and  then 
slipping  the  cards  over  each  other  uniform  amounts.  The  change 
of  the  shape  of  a  body,  or  the  strain  of  a  body,  here  illustrated,  is 
called  lamellar  motion  or  shearing  motion.  It  is  a  form  of  motion 
of  very  great  importance. 


4 

m 

\ 

0 

/  /  / " 

/  /  /a 

/ 

\ 

Jl 

A 

-3        -2 

I 

Vx 

X 

A 

!            3 

4 

-•>. 

\ 

/ 

/ 

-3 

\ 

■4 

Fig.  47. — The  shear  of  the  cubical  parabola  2/  =  a'  in  the  line  y  = 
X,  and  also  in  the  fine  y  =  —  x. 

We  shall  speak  of  the  locus  y  =  f{x)  +  mx  as  the  shear  of  the 
curve  y  =  f(x)  in  the  line  y  =  mx. 


Theorems   on  Loci 

XIII.  The  addition  of  the  term  mx  to  the  right  side  of  y  =  f{x) 
shears  the  locus  y  =  f(x)  in  the  line  y  =  mx. 
The  locus 

y  —  ax^  +  TOcc  +  6 

is  made  from  y  =  a;'  by  a  combination  of  first,  a  uniform  elongation 


THE  POWER  FUNCTION 


89 


[a],  second,,  a  shearing  motion  [m],  and  third,  a  translation  [6]. 
Either  motion  may  be  changed  in  sense  by  changing  the  sign  of 
a,  m,  or  6,  respectively. 

The  student  may  easily  show  that  the  effect  of  a  shearing  motion 
upon  the  straight  line  y  =  mx  +  b  is  merely  a  rotation  about 
the  fixed  point  (0,  b).    The  line  is  really  stretched  in  the  direction 


M 


H 

K 

8 

7 

0 
5 
4 
3 

2 

1 

- 

o 

o 

o 

0 

0 

o 

0 

2 

o 
o 

o 

I 

o 

0 

)" 

1 

2 

3 

4 

5 

6 

7 
3 

- 

1 

2 

Fig.  48. — Shearing  motion  illustrated  by  the  slipping  of  the  members 
of  pack  of  cards. 


of  its  own  length,  but  this  does  not  change  the  shape  of  the  line 
nor  does  it  change  the  line  geometrically.  A  line  segment  (that 
is,  a  hne  of  finite  length)  would  be  modified,  however. 

The  parabola  y  =  x^  is  transformed  under  a  shearing  motion 
in  a  most  interesting  way.     For,  after  shear,  y  =  x'  becomes 

y  =  x^  +  2mx,  (4) 

where,  for  convenience,  the  amount  of  the  shearing  motion  is 


90 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§38 


represented  by  2w  instead  of  by  m.    Writing. this  in  the  form 

y  =  x^  +  2mx  +  m^  —  m^, 
or 

2/  =  (x  +  my  —  m^, 

y  +  m''  =  (x  +  mY,  (5) 

we  see  that  (4)  can  be  made  from  the  parabola  y  =  x'^  hy  trans- 
lating the  curve  to  the  left  the  amount  m  and  down  the  amount  m^. 
(See  Fig.  49.) 


\ 

\ 

4 
S 

"11 

<='7  / 
"V  / 

\ 

\ 

•>. 

\\ 

-^ 

fe*/' 

■3 

-2 

T 

0 

-1 

. 

I 

3 

y^ 

-9. 

.9. 

-4 

Fig.  49. — The  shear  oi  y  =  x^  in  the  line  y   =  0 .  6x. 

Shearing  motion,  therefore,  rotates  the  straight  line  and  trans- 
lates the  parabola.  The  effect  on  other  curves  is  much  more 
complicated,  as  is  seen  from  Figs.  46  to  48. 

The  parabola  y  =  x^  after  shear  is  identical  in  size  and  shape 
with  y  =  x^  -\-  mx  +  b.  Likewise,  y  =  ax'  -\-  bx  +  c  is  a  para- 
bola differing  only  in  position  from  y  =  ax'. 

Exercises 

1.  Explain  how  the  curve  y  =  x^  -\-  2x  may  be  made  from  the 
curve  y  =  x^.  How  can  the  curve  y  =  2x'  +  3x  be  made  from  the 
curve  y  =  2x'? 


§39]  THE  POWER  FUNCTION  91 

2.  Find  the  coordinates  of  the  lowest  point  oi  y  —  x^  —  ix,  that 
is,  put  this  equation  in  the  form  y  —  b  =  {x  —  a)^. 

3.  Compare  the  curves  y  =  x'  -\-  2x  and  y  =  x^  —  2x.  (Do  not 
draw  the  curves.) 

4.  Explain  how  the  curve  y  =  1/x  +  2x  may  be  formed  from  the 

curve  y  =  1/x  and  oi  y  =  2x. 

• 

,  39.  Rotation  of  a  Locus.  The  only  simple  type  of  displace- 
ment of  a  locus  not  yet  considered  is  the  rotation  of  the  locus 
about  the  origin  0.     This  will  be  taken  up  in  the  next  chapter. 

40.  Roots  of  Functions.  The  roots,  or  zeros,  of  a  function  are 
the  values  of  the  argument  for  which  the  corresponding  value  of 
the  function  is  zero.  Thus,  2  and  3  are  rgots  of  the  function 
x^  —  5x  +  6,  for  substituting  either  number  for  x  causes  the 
function  to  become  zero.  The  roots  of  a;^  —  a;  —  6  are  +  3  and 
-  2.     The  roots  of  x^  -  6x^  +  llx  -  6  are  1,  2,  3. 

The  word  root,  used  in  this  sense,  has,  of  course,  an  entirely 
different  significance  from  the  same  word  in  "square  root,"  "cube 
root,''  etc.  But  the  roots  of  the  function  x''  —  5x  —  6  are  also  the 
roots  of  the  equation  x'  —  5x  —  Q  =  0. 

In  the  graph  of  the  cubic  function  y  =  x'  —  x  in  Fig.  47,  the 
curve  crosses  the  X-axis  at  a;  =  —  1,  x  =  0,  and  x  =  1.  These 
are  the  values  of  x  that  make  the  function  x^  —  x  zero,  and  are,  of 
course,  the  roots  of  the  function  a;'  —  x.  No  matter  what  the 
function  may  be,  it  is  obvious  that  the  intercepts  on  the  X-axis  of 
the  curve  y  =  f{x),  as  OA,  OB,  Fig.  47,  must  represent  the  roots 
off(x). 

Exercises 

1.  From  the  curve  y  =  x^  sketch  the  curves  j/  —  4  =  x^;  i/  =  4x^; 
^y  =  x^;  y  =  (x  -  4)2. 

„    ^,       ,  x'  .       ,  X*       ,  (x  —  3)2 

2.  Sketch  y  =  -i^;y  =  ^^  -  z]  V  =  -2   -  ^'<y  =  2 

ill  1 

3.  Sketch    the    curves   y  =  x^;  y  =  x^;  y  =  2x^;   y  =  (x  —  2)^; 

y  -2  =  {x  -  2)*  and  y  =  (x  -  3)*. 

4.  Sketch  the  curves  y'  =  (x-3)';  (2/  -  2)2  =  x',  and  (y  -  2)^  = 
(x  -  3)^ 

5.  Graph  yi  =  x  and  y^  =  x'  and  thence  y  =  x  +  x'. 

6.  Find  the  X-intercepts  for  the  following : 

[a)  y  =  x^  +  2x  -  3;    (b)  y  =  x'  -  x;         (c)  y  =  2x''  +  x  -  3. 


92  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§41 

7.  Find  tbe  roots  of  the  following  functions: 

(a)  x^  -Qx  +  8;  (6)  x'  +x  -2; 

(c)  x'  -  X  -  6;  (d)  2x'  -  5s  +  2; 

(e)   Qx"  +x  -  1;  (/)  x^  +  x'-  2x. 

41.  Intersectioii  of  Loci.  Any  pair  of  values  of  x  and  y  that 
satisfies  an  equation  containing  x  and  y  locates  some  point  on 
the  graph  of  that  equation.  Consequently,  any  set  of  values  of 
X  and  y  that  satisfies  both  equations  of  a  system  of  two  equations 
containing  x  and  y,  must  locate  some  point  common  to  the 
graphs  of  the  two  equations.  In  other  words,  the  coordinates 
of  a  point  of  intersection  of  two  graphs  is  a  solution  of  the  equa- 
tions of  the  graphs  considered  as  simultaneous  equations. 

To  find  the  values  of  x  and  y  that  satisfy  two  equations,  we 
solve  them  as  simultaneous  equations.  Hence,  to  find  the  points 
of  intersection  of  two  loci  we  must  solve  the  equations  of  the 
two  curves.  There  will  be  a  pair  of  values  or  a  solution  for  each 
point  of  intersection. 

Thus,  the  intersection  of  the  lines  y  =  3x  —  2  and  y  =  x/2  +  3 
is  the  point  (2,  4)  and  a;  =  2,  ?/  =  4,  is  the  solution  of  the  simul- 
taneous equations. 

Exercises 

Find  the  point  or  points  of  intersection  of  the  following  pairs  of  loci: 
l.y  —  X  —  S  and  y  =  2x  +  1. 

2.  y  =  x^  and  y  —  x. 

3.  y  =  2x^  and  y  =  ix  +  1. 

4.  2/  =  a;'  and  y  —  2x. 

5.  y  =  —  X  and  x'  +  y'  =  2Sr. 

Miscellaneous  Exercises 

1.  Find  the  slope,  the  y-intercept,  and  the  X-intercept  for  the 
following: 

Co)  y  =2x  -3;  (b)  y  =  x  +  2;  (c)  3y  -  6x  =  10. 

2.  Write  the  equations  of  the  lines  determined  by  the  following 
data: 

(o)  slope      2  F-intercept      5 

(ft)  slope  —2  y-intercept      5 

(c)  slope     2  y-intercept  —5 

(d)  slope  —2  y-intercept  —5 

(e)  slope  —2  X-intercept     4 


§41]  THE,  POWER  FUNCTION  93 

3.  Doesg  the  line  3y  —  2x  =  1  pass  through  the  point : 

(a)  (1,  1);  (6)  (2,  2);  (c)  (-2,  -1);  (d)  (0,  0);  (e)  ^3,  4). 

4.  Find  the  equation  of  a  straight  line  with  slope  2  and  passing 
through  the  point  (3,  2). 

5.  Write  the  equations  of  the  lines  determined  by  the  following 
data: 

(a)  slope  1  and  passing  through  (1,  1). 
(6)  slope  —1  and  passing  through  (  —  1,  1). 
(c)  slope  2  and  passing  through  (1,  —3). 
{d)  slope  —3  and  passing  through  (—2,  —1). 

6.  Write  the  equation  of  a  line  passing  through  (2,  1)  and  (3,  —5). 

7.  Write  the  equations  of  the  lines  passing  through  the  following 
pairs  of  points: 

(a)  a,  1)  and  (2,  3);  ^6)  (3,  -1)  and  (-2,  1); 
(c)  (2,  -3)  and  (2,  1);  {d)  (1,  -5)  and  (-2,  -3). 
(e)  (0,  2)  and  (3,  0);  (/)  (0,  0)  and  (-3,  2). 

8.  Make  two  suitable  graphs  upon  a  single  sheet  of  squared  paper 
from  the  following  data  giving  the  highest  and  lowest  average  closing 
price  of  twenty-five  leading  stocks  listed  on  the  New  York  Stock  Ex- 
change for  the  years  given  in  the  table: 


Year 

Highest 

Lowest 

1913 

94.56 

79.58 

1912 

101.40 

91.41 

1911 

101.76 

86.29 

1910 

111.12 

86.32 

1909 

112.76 

93.24 

1908 

99.04 

67.87 

1907 

109.88 

65.04 

1906 

113.82 

93.36 

1905 

109.05 

90.87 

1904 

97.73 

70.66 

1903 

98.16 

68.41 

1902 

101.88 

87.30 

Should  smooth  curves  be  drawn  through  the  points  plotted  from  this 
table? 

9.  Plot  the  data  given  in  following  table  upon  squared  paper.  Use 
the  same  horizontal  axis  for  all  three  curves.  Put  the  temperature 
curves  above  the  discharge  curve,  using  the  same  horizontal  (time) 
scale  for  both.     Let  1  cm.  on  the  vertical  scale  represent  0.1  second- 


94 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§41 


foot'  discharge,  and  10°  temperature.  Start  the  temperature  scale 
with  60°,  and  place  the  60,  6  cm.  above  the  horizontal  scale.  Start 
the  discharge  scale  with  3.4  placed  on  the  horizontal  scale. 

Discharge  op  a  Seepage  Ditch 


Time, 
Aug.  24,  1905 

Discharge  of  ditch, 
seo.-ft. 

Temp,  of  water, 

\-F. 

Temp,  of  air. 

8  :  00  a.  m. 

3.72 

65 

9:00  a.m. 

3.70 

70 

74 

11 :00  a.  m. 

3.66 

79 

84 

1:30  p.m. 

3.52 

83 

85 

2:15  p.m. 

3.49 

3:30  p.m. 

3.52 

84 

90 

5:30  p.m. 

3.66 

78 

88 

6:00  p.m. 

3.73 

8:00  p.  m. 

3.84 

67 

76 

10.  Plot  data  given  in  the  following  table.''    Plot  j^  along  the 

horizontal  axis  using  1  cm.  to  represent  one-tenth  unit.  Plot  veloci- 
ties along  the  vertical  axis,  using  1  cm.  to  represent  two  one-hun- 
dredths  of  1  foot  per  second.     Sketch  a  smooth  curve  among  the 


Relation  Between  Velocity  and  Depth  at  a  Point  in  the 
Lower  Mississippi 

Depth  at  observed  point  =  d;  whole  depth  =  D 


d 
D 

Velocity,  feet  per 
second 

d 
D 

Velocity,  feet  per 
second 

0.0 

3.195 

0.5 

3.228 

0.1 

3.230 

0.6 

3.181 

0.2 

3.253 

0,7 

3.127 

0.3 

3.261 

0.8 

3.059 

0.4 

3.252 

0.9 

2.976 

i 

1  Second-foot  (or  sec.-ft.)i  when  applied  to  the  measurement  of  flow  of  water 
means  one  cubic  foot  per  second. 

2  The  velocity  at  any  point  of  a  moving  stream  is  determined  by  a  current  meter 
placed  at  that  point. 


§41]  THE  POWER  FUNCTION  '  95 

points.  The  curve  may  not  pass  through  all  the  plotted  points.' 
Begin  to  number  the  vertical  axis  with  2.90.  The  true  origin  will  be 
far  below  the  sheet  of  paper. 

11.  Write  the  equations  of  the  following  curves  after  translated, 
two  units  to  the  right;  three  units  to  the  left;  five  units  up;  one  unit 
down;  two  units  to  the  left  and  one  unit  down: 

(a)  y  =  2x\  (6)  y  =  -Zx';  (c)  y  =  x^;  {d)  y  =- 

X 
1  1  3  2 

W2/=^;    U)y=~^;       (,g)  y  =  x^-,  (h)  y  =  x^. 

Sketch  each  curve  in  its  original  and  also  in  its  translated  position. 

12.  Write  the  equation  of  each  curve  of  exercise  11  when  re- 
flected in  the  X-axis;  in  the  K-axis;  in  the  line  y  =  x.  Sketch  each 
curve  before  and  after  reflection. 

13.  Shear  each  curve  of  exercise  11  in  the  line  y  =  ix;in  the  line 
y  =  —  I  x;  in  the  line  y  =  x;  in  the  line  y  =  —  x.  Sketch  each  curve 
in  its  original  and  sheared  position. 

14.  Draw  on  a  sheet  of  coordinate  paper  the  lines  z  =  0,  x  =  1, 
x  =  —1,  y  =  0,  y  =  1,  y  =  —1.  Shade  the  regions  in  which  the 
hyperbolic  curves  lie  with  vertical  strokes;  and  those  in  which  the 
parabolic  curves  lie  with  horizontal  strokes.  Write  down  all  that 
the  resulting  figure  tells  you. 

15.  Consider  the  following:  y  =  x^,  y  =  x~',  y  =  xi,  xy  =  —  1, 
y  =  —x^,y^  =  X*,  y'  =  x%  xy  =  I,  x^  =  —  y',  x*  =  —  y'.  In  which 
equation  is  y  an  increasing  function  of  x  in  the  first  quadrant?  For 
which  does  the  slope  of  the  curve  increase  in  the  first  quadrant?  For 
which  does  the  slope  of  the  curve  decrease  in  the  first  quadrant? 

16.  Which  of  the  curves  of  exercise  11  pass  through  (0,  0)?  Through 
(1,  1)?     Through  (-1,  -1)? 

17.  Find  the  vertex  of  the  curve  y  =  x^  —  24x  -|-  150. 

Note  :  The  lowest  point  of  the  parabola  y  =  x^  may  be  called  the 
vertex. 

Suggestion  :  It  is  necessary  to  put  the  equation  in  the  form  y  —  b 
.  =(x  —  ay.  This  can  be  done  as  follows:  Add  and  subtract  144  on 
the  right  side  of  the  equation,  obtaining 

2/  =  x^  -  24x  +  144  -  144  +  150, 

1  This  curve  is  called  »  vertical  velocity  curve.  In  practical  -  work,  however, 
velocities  are  plotted  along  the  horizontal  axis  and  depths  along  the  vertical  axis, 
and  down  from  the  origin.  Your  drawing  gives  the  usual  form  of  plotting  if  it  is 
turned  90"  in  a  clockwise  direction.  Vertical  velocity  curves  are  parabolic  in  shape, 
with  the  axis  of  the  parabola  parallel  to  the  surface  of  the  water.  . 


96  ELEMENTARY  MATHEMATICAL  ANALYSIS        [|41 


3/  «  (s  -  12)»  +  6, 
or 

y  -Q  =  {x  -  12)'. 

Then  this  is  the  carve  y  =  x^  translated  12  units  to  the  right  and  6 
units  up.  -  Since  the  vertex  oi  y  =  x^  is  at  the  origin,  the  vertex  of  the 
given  curve  must  be  at  the  point  (12,  6). 

18.  Find  the  vertex  of  the  parabola  V  =  x'  —  6x  +  11. 

19.  Find  the  vertex  of  ^  =  x'  +  8x  +  1. 

20.  Find  the  vertex  oi  4  +  y  =  x'  —  7x. 

21.  Find  the  vertex  of  ^  =  9x'  +  18a;  +  1. 

22.  Translate  y  =  4a;'  —  12a;  +  2  so  that  the  equation  will  have 
tke  form  y  =  4x'. 

23.  Does  the  line  2y  ==  5x  —  1  pass  through  the  point  of  intersec- 
tion of  y  =  2a;  +  1  and  y  —  3x  —  2t 


CHAPTER  IV 


THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS 

42.  Equation  of  the  Circle.  In  rectangular  coordinates  the 
abscissa  x,  and  the  ordinate  %,  of  any  point  P  (as  OD  and  DP, 
Fig.  50)  form  two  sides  of  a  right  triangle  whose  hypotenuse 
squared  is  a;^  +  j/'.  If  the  point  P  move  in  such  manner  that  the 
length  of  this  hypotenuse  remains  , 

fixed,  the  point  P  describes  a  -  '^ 

circle  whose  center  is  the  origin 
(Fig.  50).  The  equation  of  this 
circle  is  therefore 

x2  +  y2  =  aS  (1)   ^, 

where  a  =  OP,  the  radius  of  the 
circle. 

It  is  sometimes  convenient  to 
write  the  equation  of  the  circle, 
solved  for  y,  in  the  form 

y  =  +  Va^  -  xK        (2) 

This  gives,  for  each  value  of  x,  the  two  corresponding  equal  and 
opposite  ordinates. 

To  translate  the  circle  of  radius  a  so  that  its  center  shall  be  at  the 
point  (h,  k),  it  is  merely  necessary  to  write 

(x  -  h)'' +  (y  -  k)^  =  a^.  (3) 

This  is  the  general  equation  of  any  circle  in  the  plane  XY,  for  it 
locates  the  center  at  any  desired  point,  {h,  k) ,  and  provides  for  any 
desired  radius  a. 

Exercises 

1.  Write  the  equations  of  the  circles  with  center  at  the  origin  having 
radii  3,  4,  11,  V2  respectively. 

2.  Write  the  equation  of  each  circle  described  in  exercise  1  in  the 
form  y  =  +  s/a^  —  x^. 

7  97 


Fig.  50. — The  circle. 


98  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§44 

3.  Which  of  the  following  points  Jie  on  the  circle  s^  +  j/^  =  169: 
(5,  12),  (0,  13J,  C-12,  5),  (10,  8),  i9,  9),  C9,  10)? 

4.  Which  of  the  following  points  lie  inside  and  which  lie  outside  of 
the  circle  x^  +  y'  =  100:  ^7,  7),  (10,  0;,  (7,  8),  (6,  8),  (-5,  9), 
(-7,  -8),  (2,  3),  (10,  5),  (\/40,  VSO),  (vlg,  9)? 

43.  The  Equation,  x^  +  y^  +  2gx  +  2fy  +  c  =  o  (1) 
may  be  put  in  the  form  (3)  §42.    For  it  may  be  written 

x^  +  2gx  +  g^  +  y^  +  2fy+P  =  g^+f-c 
or 

(x  +  gy  +  {v+  f)'  ^  (Vg'+P-'c  y,  (2) 

which  represents  a  circle  of  radius  -v/jf"  +P  —  c  whose  center  is  at 
the  point  (.— g,  — /).  In  case  g^  +  P  —  c  <.0,  the  radical 
becomes  imaginary,  and  the  locus  is  not  a  real  circle;  that  is, 
coordinates  of  no  points  in  the  plane  XY  satisfy  the  equation.  If 
the  radical  be  zero,  the  locus  is  a  single  point. 

44.  Any  equation  of  the  second  degree,  in  two  variables,  lacking 
the  term  xy  and  having  like  coefficients  in  the  terms  x^  and  y^,  repre- 
sents a  circle,  real,  null  or  imaginary.  The  general  equation  of 
the  second  degree  in  two  variables  may  be  written 

ax"  +  by'  +  2hxy  +  2gx  +  2fy  +  c  =  0.  (3) 

For,  when  only  two  variables  are  present,  there  can  be  present  three 
terms  of  the  second  degree,  two  terms  of  the  first  degree,  and  one 
term  of  the  zeroth  degree.  When  a  =  b  and  h  =  0  this  reduces 
to  the  form  of  (1)  above  after  dividing  through  by  a. 

Exercises 

Find  the  centers  and  the  radii  of  the  circles  given  by  the  following 
equations: 

1.  x'  +  y'  =  25.  Also  determine  which  of  the  following  points  are 
on  this  circle:  (3,  4),  (5,  5),  (4,  3),  (-3,  -4;,  (-3,  4),  (5,  0),  (2,  V2i). 

2.  x'  +  2/2  =16.  4.  x^  +  y'  -36  =  0. 

3.  x'+y'-  -i  =  0.  6.  X'  +y''  +2x  =  0. 

B.  y  =  ±  -\/l69  —  x'.  Also  find  the  slope  of  the  diameter  through 
the  point  (5,  12).     Find  the  slope  of  the  tangent  at  (5,  12). 


§45]      THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS       99 

7.  9  -  a;2  -  2/2  =  0.  10.  (x  +  a)'  +  (y  -  b)'  =  50. 

8.  x^  +y'  -Qy  =  16.  11.  x'  +  y^  +  6x  -  2y  ^  10. 

9.  x^  -2x  +y^  -  &y  =  16..       12.  x''  +  y'  -  ix  +  6y  =  12. 

13.  x2  +  j/2  -  4x  -  82/  +  4  =  0. 

14.  3x'  +  3y'  +  6x  +  12y  -  60  =  0, 

16.  Is  x^  +  2y^  +3x  —  4:y  —  12  =  Q  the  equation  of  a  circle? 
Why? 

16.  Is  2x'  +  2y'  —  3x  +  4y  —  8  =  0  the  equation  of  a  circle? 
Why? 

45.  Angular  Magnitude.    By  tHe  magnitude  of  an  angle  is 

meant  the  amount  of  rotation  of  a  line  about  a  fixed  point.    If 
a  line  OA  rotate  in  the  plane  XY  about  the  fixed  point  0  to  the 


Fig.  51. — Positive  angles. 


Fig.  52. — Negative  angles. 


position  OP,  the  line  OA  is  called  the  initial  side  and  the  hne  OP  is 
called  the  terminal  side  of  the  angle  AOP.  The  notion  of  angular 
magnitude  as  introduced  in  this  definition  is  more  general  than 
that  used  in  elementary  geometry.  There  are  two  new  and  very 
important  consequences  that  follow  therefrom: 

(1)  Angular  magnitude  is  unlimited  in  respect  to  size — that  is, 
it  may  be  of  any  amount  whatsoever.  An  angular  magnitude 
of  100  right  angles  or  twenty-five  complete  rotations  is  quite 
as  possible,  under  the  present  definition,  as  an  angle  of  smaller 
amount. 

(2)  Angular  magnitude  exists,  under  the  definition,  in  two 
opposite  senses — for  rotation  may  be  clockwise  or  anti-clockwise. 


100        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§46 

As  is  usual  in  mathematics,  the  two  opposite  senses  are  distin- 
guished by  the  terms  positive  and  negative.  If  the  rotation  is 
anti-clockwise  the  angle  is  positive;  if  clockwise  it  is  negative. 
In  Fig.  51,  AOPi,  AOPi,  AOP3,  and  AOPt  are  positive  angles. 
In  designating  an  angle  its  ihitial  side  is  always  named  first.  Thus, 
in  Fig.  51,  AOPi  designates  a  positive  angle  of  initial  side  OA. 
In  Fig.  52,  AOPi,  AOPi,  AOP3,  and  AOPt  are  negative  angles. 

In  Cartesian  coordinates,  OX  is  usually  taken  as  the  initial 
line  for  the  generation  of  angles.  If  the  terminal  side  of  an 
angle  falls  within  the  first  quadrant,  the  angle  is  said  to  be  an 
angle  of  the  first  quadrant.  If  the  terminal  side  of  any 
angle  falls  within  the  second  quadrant,  it  is  said  to  be  an  "angle 
of    the    second    quadrant,"    etc. 

Two  angles  which  differ  by  any  multiple  of  360°  are  called 
congruent  angles.  We  shall  find  that  in  certain  cases  congruent 
angles  may  be  substituted  for  each  other  without  modifying  results. 

The  theorem  in  elementary  geometry,  that  angles  at  the 
center  of  a  circle  are  proportional  to  the  intercepted  arcs,  holds 
obviously  for  the  more  general  notion  of  angular  magnitude  here 
introduced. 

46.  Units  of  Measure.  Angular  magnitude,  like  all  other 
magnitudes,  must  be  measured  by  the  application  of  a  suitable 
unit  of  measure.     Four  systems  are  in  common  use: 

(1)  Right  Angle  System.  Here  the  unit  of  measure  is  the  right 
angle,  and  all  angles  are  given  by  the  number  of  right  angles  and 
fraction  of  a  right  angle  therein  contained.  This  unit  is  famihar 
to  the  student  from  elementary  geometry.  A  practical  illus- 
tration is  the  scale  of  a  mariner's  compass,  in  which  the  right  angles 
are  divided  into  halves,  quarters  and  eighths. 

(2)  The  Degree  System.  Here  the  unit  is  the  angle  corre- 
sponding to  xriT  of  a  complete  rotation.  This  system,  with  the 
sexagesimal  sub-division=  (division  by  60ths)  •  into  minutes 
and  seconds,  is  familiar  to  the  student.  This  system  dates  back 
to  remote  antiquity.  It  was  used  by,  if  it  did  not  originate 
among,  the  Babylonians. 

(3)  The  Hour  System.  In  astronomy,  the  angular  magnitude 
about  a  point  is  divided  into  24  hours,  and  these  into  minutes 


THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS     101 


and  seconds.    This  system  is  analogous  to  our  system  of  measuring 
time. 

(4)  The  Radian,  or  Circular  System.  Here  the  unit  of  measure 
is  an  angle  such  that  the  length  of  the  arc  of  a  circle  described  about 
the  vertex  as  center  is  equal  to  the  length  of  the  radius  of  the 
circle.  This  system  of  angular  measure  is  fundamental  in  me- 
chanics, mathematical  physics  and  pure  mathematics.  It  must 
be  thoroughly  mastered  by  the  student.  The  unit  of  measure  in 
this  system  is  called  the  radian.     Its  size  is  shown  in  Fig.  53. 


O  Radius 

Fig.  53. — Definition  of  the  Radian.     The  angle  AOP  is  one  Radian. 


Inasmuch  as  the  radius  is  contained  2ir  times  in  a  circumference, 
we  have  the  equivalents: 

2%  radians  =  360°. 
or  1  radian  =  57°  17'  44".8  =  57°  17'.7  =  57°.3  nearly. 

1  degree  =  0.01745  radians. 

The  following  equivalents  are  of  special  importance: 
a  straight  angle  =  x  radians. 

a  right  angle  =  ^  radians. 

60°  =  ^  radians. 
o 


45° 


radians. 


30°  =  ^  radians, 
o 


102        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§47 

There  is  no  generally  adopted  scheme  for  writing  angular  magni- 
tude in  radian  measure.  We  shall  use  the  superior  Roman  letter 
"'"  to  indicate  the  measure.     For  example,  18°  =  O.SMW. 

Since  the  circumference  of  a  circle  is  incommensurable  with  its 
diameter,  it  follows  that  the  number  of  radians  in  an  angle  is 
always  incommensurable  with  the  number  of  degrees  in  the  angle. 

The  speed,  or  angular  velocity,  of  rotating  parts  is  usually 
given  either  in  revolutions  per  minute  (abbreviated  "r.p.m.") 
or  in  radians  per  second. 

47.  Uniform  Circular  Motion.  Suppose  the  line  OP,  Fig.  50, 
is  revolving  counter-clockwise  at  the  rate  of  ¥  per  second,  the 
angle  AOP  in  radians  is  then  ht,  t  being  the  time  in  seconds  re- 
quired for  OP  to  turn  from  the  initial  position  OA.  If  we  call  d 
the  angle  AOP,  we  have  B  =  kt&s  the  equation  defim'ng  the  motion. 
The  following  terms  are  in  common  use: 

1.  The  angular  velocity  of  the  uniform  circular  motion  's  k 
(radians  per  second). 

2.  The  amplitude  of  the  uniform  circular  motion  is  a. 

3.  The  period  of  the  uniform  circular  motion  is  the  number  of 
seconds  required  for  one  revolution. 

4.  The  frequency  of  the  uniform  circular  motion  is  the  number 
of  revolutions  per  second. 

Sometimes  the  unit  of  time  is  taken  as  one  minute.  Also  the 
motion  is  sometimes  clockwise,  or  negative. 

Exercises 

1.  Express  each  of  the  following  in  radians:  135°,  330°,  225°,  15°, 
150°,  75°,  120°.     (Do  not  work  out  in  decimals;  use  jr). 

2.  Express  each  of  the  following  in  degrees:  0.2'',  ^u^'  •fTr'''  ^ir'- 

3.  How  many  revolutions  per  minute  is  20  radians  per  second? 

4.  The  angular  velocity,  in  radians  per  second,  of  a  36-inch  auto- 
mobile tire  is  required,  when  the  car  is  making  20  miles'per  hour. 

6.  What  is  the  angular  velocity  in  radians  per  second  of  a  6-foot 
drive-wheel,  when  the  speed  of  the  locomotive  is  50  miles  per  hour? 

6.  The  frequency  of  a  cream  separator  is  6800  r.p.m.  What  is 
its  period,  and  its  angular  velocity  in  radians  per  second? 

7.  A  wheel  is  revolving  uniformly  30^  per  second.  What  is  its  per- 
iod and  frequency? 


§48]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   103 

8.  The  speed  of  the  turbine  wheel  of  a  5-h.p.  DeLaval  steam  turbine 
is  30,000  r.p.m.     What  is  the  angular  velocity  in  radians  per  second? 

48.  The  Circular,  or  Trigonometric  Functions.  To  each  point 
on  the  circle  x^  +  y^  =  a?  there  corresponds  not  only  an  abscissa 
and  an  ordinate,  but  also  an  angle  6  <  360°,  as  shown  in  Fig.  50. 
This  angle  is  called  the  direction  angle,  or  vectorial  angle,  of  the 
point  P.  When  9  is  given,  x,  y,  and  a  are  not  determined,  but  the 
ratios  y/a,  x/a,  y/x,  and  their  reciprocals,  a/y,  a/x,  x/y  are  de- 
termined. Hence  these  ratios  are,  by  definition(§6),  fimctions 
of  d.  They  are  known  as  the  circular,  or  trigonometric,  functions 
of  6,  and  are  named  and  written  as  follows: 


Function  of  e 

Name 

Written 

y/a. 

sine  of  0. 

sin  0. 

x/a. 

cosine  of  0. 

cos  0. 

y/x. 

tangent  of  0. 

tan  0. 

x/y. 

cotangent  of  0. 

cots. 

a/x. 

secant  of  0. 

sec0. 

a/y. 

cosecant  of  0. 

CSC  0. 

The  circular  functions  are  usually  thought  of  in  the  above  order; 
that  is,  in  such  order  that  the  first  and  last,  the  middle  two,  and 
those  intermediate  to  these,  are  reciprocals  of  each  other. 

The  names  of  the  six  ratios  must  be  committed  to  memory. 
They  should  be  committed,  using  the  names  of  x,  y,  and  a  as 
follows : 

Ratio  Written 

ordinate/radius.  sin  0. 

abscissa/radius.  cos  0. 

ordinate/abscissa.  tan  0. 

abscissa/ordinate.  '                 cot  0. 

radius/abscissa.  sec  0. 

radius/ordinate.  esc  0. 

The  right  triangle  DOP,  Fig.  50,  of  sides  x,  y,  and  a,  whose  ratios 
give  the  functions  of  the  angle  XOP,  is  often  called  the  triangle  of 
reference  for  this  angle.  It  is  obvious  that  the  size  of  the  triangle 
of  reference  has  no  eifect  of  itself  upon  the  value  of  the  functions 
of  the  angle.    Thus  in  Fig.  51  either  DiOPi  or  D/OPi'  may  be 


104        ELEMENTARY  MATHEMATICAL  ANALYSIS 


taken  as  the  triangle  of  reference  for  the  angle  6.    Smce  the 
triangles  are  similar  we  have 

P^D,     P.'Di'  P^D,      Pi'D,' 


ODi        OD,' 


OPi        OPi' 


etc.,  which  shows  that  identical  ratios  or  trigonometric  functions  of 
6  are  derived  from  the  two  triangles  of  reference. 

49.  1  Elaborate  means  for  computing  the  six  functions  have  been 
devised  and  the  values  of  the  functions  have  been  placed  in 
convenient  tables  for  use.  The  functions  are  usually  printed  to 
3,  4,  5  or  6  decimal  places,  but  tables  of  8,  10  and  even  14  places 
exist.    The  functions  of  only  a  few  angles  can  be  computed  by 


o^V? 


i'lG.  54. — Triangles  of  reference  for  angles  of  30°,  45°,  and  60° 


elementary  means;  these  angles  are,  however,  especially  important. 
(1)  The  Functions  of  30°.     In  Fig.  54a,  if  angle  AOB  be  30°, 
angle  ABO  must  be  60°.     By  constructing  the  equilateral  triangle 
BOB',  each  angle  of  triangle  BOB'  will  be  60°,  and 

y  =  AB  =  ^BB'  =  ia. 
Therefore  ' 


sin  30° 


Also, 


OA  =  \0B^  -  AB^  =  Va^  -  ia^  =  iaVs. 


Therefore 

sin  30°  =  i, 

1  Some  will  prefer  to  take  §50  before  §49. 


§49]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS  105 

o  2  ' 


tan  30°  =  ^^-  =  ^, 

1  30°      ^  ^' 
1  2V3 


•=°*3«°  =  t^  =  ^/3, 


sec  30°  = 


cos  30°         3    ' 

CSC  30°  =  ^^  =  2. 
sm  30 

(2)  Functions  of  45°.    In  the  diagram,  Fig.  546,  the  triangle 
AOB  is  isosceles,  or  y  =  a;,  and  a^  =  x^  +  y'  =  2x^.    It  follows 
that  a  =  X  •\/2  =  y  -\/2- 
Therefore 

sin45°  =  ^=^, 
2/  \/2        2  \ 


cos  45° 

a;          V2 

xV2        2  ' 

tan  45° 

=  '-  =  !,- 

X 

cot  45° 

^           1 

tan  45°       ' 

sec  45° 

~  cos  45°  ~  ^^' 

CSC  45° 

-sin  45°  =^2. 

(3)  Functions  of  60°.    In  the  diagram.  Fig.  54c,  construct  the 

equiangular  triangle  B'OB;  then  it  is  seen  that,  as  in  case  (1) 

above, 

OA  =  h  OB'  =  h  a- 
and 

y  =  VaT^^^Ya^  =  i  a\/3. 
Therefore 

sin  60°  =  *"^^  ^  V3 
a  2 


106        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§50 


cos  60°  =  ^  =  i 

tan  60°-^''^  = 

50 

=  V3. 

cot  60°  — 

_  V3 

tan  60° 

3 

r-nr   Rf\° 

=  2. 

COS  60° 

CSC  60°  -       ^ 

.  2V3 

sin  60° 

50.  Graphical  Computation  of  Circular  Functions.  Approxi- 
mate determination  of  the  numerical  values  of  the  circular  func- 
tions of  any  given  angle  may  be  made  graphically  on  ordinary 
coordinate  paper.  Locate  the  vertex  of  the  angle  at  the  inter- 
section of  any  two  lines  of  the  squared  paper,  form  Ml.  Let 
the  initial  side  of  the  angle  coincide  with  one  of  the  rulings  of  the 
squared  paper  and  lay  off  the  terminal  side  of  the  angle  by  means 
of  a  protractor.  If  the  sine  or  cosine  is  desired,  describe  a  circle 
about  the  vertex  of  the  angle  as  center  using  a  radius  appro- 
priate to  the  scale  of  the  squared  paper — for  example,  a  radius  of 
10  cm.  on  coordinate  paper  ruled  in  centimeters  and  fifths  (form 
Ml)  permits  direct  reading  to  j-^  of  the  radius  a  and,  by  interpo- 
lation, to  j^u"  of  the  radius  a.  The  ordinate  and  abscissa  of  the 
point  of  intersection  of  the  terminal  side  of  the  angle  and  the  circle 
may.  then  be  read  and  the  numerical  value  of  sine  and  cosine  com- 
puted by  dividing  each  of  these  by  the  length  of  the  radius. 

If  the  numerical  value  of  the  tangent  or  cotangent  be  required, 
the  construction  of  a  circle  is  not  necessary.  The  angle  should 
be  laid  off  as  above  described,  and  a  triangle  of  reference  con- 
structed. To  avoid  long  division,  the  abscissa  of  the  triangle  of 
reference  may  be  taken  equal  to  50  or  100  mm.  for  the  determina- 
tion of  the  tangent;  and  the  ordinate  may  be  taken  equal  to  50 
or  100  mm.  for  the  determination  of  the  cotangent. 

The  following  table  (Table  III)  contains  the  trigonometric 
functions  of  acute  angles  for  increments  of  10°  of  the  argument 
from  0°  to  90°.     (See  the  end  of  the  book  for  a  larger  table.) 


§50]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   107 

Table  III 
Natural  Trigonometric  Functions  to  Two  Decimal  Places 


0'. 

9' 

sin  d 

cos  0 

tan  e 

cot  0 

sec  8 

CSC  d' 

0 

0.00 

0.00 

1.00 

0.00 

00 

1.00 

03 

10 

0.17 

0.17 

0.98 

0.18 

5.67 

1.02 

5.76 

20 

0.35 

0.34 

0.94 

0.36 

2.75 

1.06 

2.92 

30 

0.52 

0.50 

0.87 

0.58 

1.73 

1.15 

2.00 

40 

0.70 

0.64 

0.77 

0.84 

1.19 

1.31 

1.56 

50 

0.87 

0.77 

0.64 

1.19 

0.84 

1.56 

1.31 

60 

1.05 

0.87 

0.50 

1.73 

0.58 

2.00 

1.15 

70 

1.22 

0.94 

0.34 

2.75 

0.36 

2.92 

1.06 

80 

1.40 

0.98 

0.17 

5.67 

0.18 

5.76 

1.02 

90 

1.57 

1.00 

0.00 

CO 

0,00 

OO 

1.00 

The  most  important  of  these  results  are  placed  in  the  following 
table : 


0° 

30° 

45° 

60° 

90° 

Sine 

0 

1 

2 

V2 
2 

V3 
2 

1 

Cosine 

1 

V3 

2 

V2 
2 

1 

0 

Tangent. . 

0 

V3 
3 

1 

V3 

CO 

V2  = 

1.4142 

V3- 

=  1.7321 

Exercises 

1.  Find  by  graphical  construction  all  the  functions  of  15°. 
Note. — A  protractor  is  not  needed  as  angles  of  45°  and  30°  may  be 

constructed  with  ruler  and  compass. 

2.  Find  tan  60°.     Compare  with  the  value  found  above  in  §  49 . 

3.  Lay  off  angles  of  10°,  20°,  30°,  and  40°  with  a  protractor  and 
determine  graphically  the  sine  of  each  angle;  record  the  results  in  a 
suitable  table. 

4.  Find  the  sine,  cosine,  and  tangent  of  75°. 
6.  Which  is  greater,  sec  40°  or  esc  60°? 

6.  Determine  the  angle  whose  tangent  is  \. 

7.  Find  the  angle  whose  sine  is  0.6. 


108        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§S1 


8.  Which  is  greater,  sin  40°  or  2  sin  20°? 

9.  Does  an  angle  exist  whose  tangent  is  1,000,000? 

61.  Signs  of  the  Functions.  The  circular  functions  have,  of 
course,  the  algebraic  signs  of  the  ratios  that  define  them.  Of  the 
three  numbers  entering  these  ratios,  the  distance,  or  radius 
a,  is  always  to  be  taken  as  positive.  It  enters  the  ratios,  there- 
fore, always  as  a  positive  number.  The  abscissa  and  the  ordinate, 
X  and  y,  have  the  algebraic  signs  appropriate  to  the  quadrants  in 
which  P  falls.  The  student  should  determine  the  signs  of  the 
functions  in  each  quadrant,  as  follows:  (See  Fig.  50.) 


First 
quadrant 

Second 
quadrant 

Third 
quadrant 

Fourth 
quadrant 

Sine 

+ 
+ 
+ 

+ 

+ 

+ 

Cosine      

Tangent 

Of  course  the  reciprocals  have  the  same  signs  as  the  original 
functions. 

The  signs  are  readily  remembered  by  the  following  scheme: 

Sine  Cosine  Tangent 


+ 


+ 


+ 


+ 


+ 


Cosecant 


Secant 


+ 


Cotangent 


52.  Triangles  of  reference,  geometrically  similar  to  those  in 
Fig.  54  for  angles  of  30°,  45°,  and  60°,  exist  in  each  of  the  four 
quadrants,  namely,  when  the  hypotenuse  of  the  triangle  of 
reference  in  these  quadrants  is  either  parallel  or  perpendicular 
to  the  hypotenuse  of  the  triangle  in  the  first  quadrant.  Then 
an  acute  angle  of  one  must  equal  an  acute  angle  of  the  other 
and  the  triangles  must  be  similar.  The  numerical  values  of  the 
functions  in  the  two  quadrants  are  therefore  the  same.  The 
algebraic  signs  are  determined  by  properly  taking  account  of 


§53]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   109 

the  signs  of  the  abscissa  and  the  ordinate  in  that  quadrant. 
Thus  the  triangle  of  reference  for  120°  is  geometrically  similar 

to  that  for  60°.    Hence,  sin  120°  = 

and  tan  120°  =  -  Vs. 

Exercises 


V3 


,  but  cos  120°  =  -  i, 


1.  The  student  wiU  fill  in  the  blanks  in  the  following  table  with 
the  correct  numerical  value  and  the  correct  sign  of  each  function: 


Function 

120° 

135° 

150° 

210° 

225° 

240° 

300° 

315° 

330° 

Sin 



Cos 

Tan 

Cot 

Sec 

Csc 

2.  Write  down  the  functions  of  390°  and  405°. 

3.  The  tangent  of  an  angle  is  1.     What  angle  <  360°  may  it  be? 

4.  Cose  =  —  |.     What  two  angles  <  360°  satisfy  the  equation? 

5.  Sec  0  =  2.     Solve  for  all  angles  <360°. 

6.  ^Csc  e  =  -  y/%     Solve  for  B  <  360°. 

53.  Functions  of  0°  and  90°.  In  Fig.  50,  let  the  angle  AOP 
decrease  toward  zero,  the  point  P  remaining  on  the  circumference 
of  radius  a.  Then  y  or  PD  decreases  toward  zero.  Therefore, 
sin  0°  =  0.  Also  x  or  OD  increases  toward  the  value  a  so  that 
the  ratio  x/a  becomes  unity,  or  cos  0°  =  1.  Likewise  the  ratio 
y/x  becomes  zero,  or  tan  0°  =  0. 

The  reciprocals  of  these  functions  change  as  follows:  As  the 
angle  AOP  approaches  zero,  the  ratio  a/y  increases  in  value 
without  limit,  or  the  cosecant  becomes  infinite.  In  symbols 
(see  §24)  csc  0°  =^  w ,    Likewise)  cot  0°  =5  «> ,  but  sec  0°  =  L 


110        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§54 


In  a  similar  way  the  functions  of  90°  may  be  investigated.  Tiie 
following  table  gives  the  variation  of  the  functions  as  the  angle 
varies  from  0°  to  90°,  from  90°  to  180°,  etc.: 


Angle 

From 
0°  to  90° 

From 
90°  to  180° 

Prom 
180°  to  270° 

From 
270°  to  360° 

Sin 

Cos 

Tan 

Cot 

Sec 

Oto  +   1 

+  Ito        0 

0  to  +«> 

+  0O   to           0 
+    1  to  +  00 
+  CO   to  +    1 

+  Ito        0 
Oto  -   1 

-o=  to        0 

0  to  -  oo 

—  oo    to    —     1 

+     1  to   +  oo 

Csc 

The  student  is  to  supply  the  results  for  the  last  two  coltimns. 

54.  Fundamental  Selations.  The  trigonometric  functions  are 
not  independent  of  each  other.  Because  of  the  relation  x^  +  j/^ 
=  a'-,  it  is  possible  to  compute  the  numerical  or  absolute  values  of 
the  remaining  five  functions  when  the  value  of  any  one  of  the  six 
is  given.  This  may  be  accompHshed  by  means  of  the  fundamental 
formulas  derived  below. 

Divide  the  members  of  the  equation 


by  a?     Then 


or, 


a:^  +  2/2  = 


1, 


sin"  e  +  cos2  e  =  I. 
Likewise  divide  (1)  through  by  x^;  then 

'  +  ©"=©" 

or 

sec2  0  =  1+  tan2  B. 

Also  divide  (1)  through  by  y'^;  then 

cgc^  ?  =  1  +  cot=  9. 


or 


(1) 


(2) 


(3) 


(4) 


THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    111 


Also,  since 


we  obtain 


and  likewise 


y 

a 

X 

a 
tan  e  = 

cot  e  = 


y 


sin  6 


cos  d 
cos  6 


(5) 


(6) 


Formulas  (2)  to  (6)  are  the  fundamental  relations  between  the  six 
trigonometric  functions.  They  must  be  committed  to  memory 
by  the  student. 


=  tan  csc-=l  +  cot- 

FiG.  55. — Diagram  of  the  relations  between  the  six  circular  functions. 


sin 
cos 


The  above  relations  between  the  functions  may  be  illustrated 
by  a  diagram  as  in  Fig.  55.  The  simpler,  or  reciprocal,  relations 
are  shown  by  the  connecting  lines  drawn  above  the  functions. 

The  reciprocal  equations  and  the  formulas  (2),  (3),  and  (4)  are 
sufficient  to  express  the  absolute  or  numerical  value  of  any  function 
of  any  angle  in  terms  of  any  other  function  of  that  angle.  The 
algebraic  sign  to  be  given  the  result  must  be  properly  selected 
in  each  case  according  to  the  quadrant  in  which  the  angle  lies. 

Exercises 

All  angles  in  the  following  exercises  are  supposed  to  be  less  than 
ninety  degrees. 


112        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§54 

1.  Sin  e  =  1/5.  Find  the  values  for  the  other  five  circular 
functions. 

Draw  a  right  triangle  whose  hypotenuse  is  5,  whose  altitude  is  1 
and  whose  base  coincides  with  OX.  In  other  words,  make  a  =  5 
and  y  =  1  in  Fig.  56.  Calculate  x  =  v'25  —  1  =  2  \/6  and  write 
down  all  of  the  functions  from  their  definitions. 

2.  Cos  9  =  1/3.     Find  esc  9. 

Take  a  =  3  and  a;  =  1  in  Fig.  56.  Find  y  and  then  write  down  the 
function  from  its  definition. 

3.  Tan  9  =  2.     Find  sin  d. 

Take  x  =  I  and  y  =  2  in  Fig.  56,  calculate  a,  and  then  write 
down  the  function  from  its  definition. 


O  X  A 

Fig.  56. — Triangle  of  reference  for  B  and  for  complement  of  S. 

4.  Sec  e  =  10.     Find  esc  6. 

Take  a  =  10  and  x  =  1  and  compute  y. 

6.  Find  the  values  of  all  functions  of  9  if  cot  6  =  1.5. 

6.  Find  the  functions  of  9  if  cos  8  =  0.1. 

7.  Find  the  values  of  each  of  the  remaining  circular  functions  in 
each  of  the  following  cases: 

(a)  sin  e  =  5/13.  (d)  tan  e  =  3/4. 


(6)   cos  e  =  4/5. 


(e)  sec  9  =  2. 


(g)  tan  6  =  m. 
{h)  sin  e  = 


Vc"  +  d' 


(c)  sec  0  =  1.25.  (/)  tan  e  =  1/3. 

Show  that  the  following  equalities  are  correct: 

8.  tan  d  cos  9  =  sin  6. 

9.  sin  e  cot  B  sec  9  =  1. 

10.  (sin  9  +  cos  9)2  =  2  sin  9  cos  9  +  1. 

11.  tan  9  +  cot  9  =  sec  9  esc  9. 

12.  Express  each  trigonometric  function  in  terms  of  each  of  the 
others;  i.e.,  fill  in  all  blank  spaces  in  the  following  table; 


§54]   THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS     113 


sin 

cos 

tan 

cot 

sec 

cso 

sin 

sin 

1 

esc 

cos 

cos 

1 
sec 

tan 

tan 

1 

cot 

cot 

1 

tan 

cot 

sec 

1 
cos 

sec 

CSC 

1 
sin 

CSC 

Fig.  57. — Diagram  for  exercise  13. 


The  following  exercises  refer  to  angles  <360°  of  any  quadrant: 
13.  If  sin  9  =  —  3/4  and  tan  B  is  positive,  find  the  remaining  five 

functions. 

Hint:  Since  sin  e  is  negative  and  tan  9  is  positive,  the  angle  9  is  in 

the  third  quadrant.    See  Fig.  57. 
8 


114        EaiEMENTARY  MATHEMATICAL  ANALYSIS        [§55 

14.  If  C08  9  =  12/13  and  sin  9  ia  negative,  find  the  remaining  five 
functions  of  6. 

15.  If  tan  e  =  —  \/3  and  cos  B  is  negative,  find  the  remaining  func- 
tions of  e. 

16.  If  cos  9  =    —  1/3  and  sin  9  is  positive,  find  the  remaining 
functions. 

17.  If  tan  9  =  5/12  and   sec  9  is  negative,  find   the  remaining 
functions  of  9. 

18.  If  sin  9  =  3/5  and  tan  9  is  negative,  find  the  jemaining  func- 
tions of  9. 


Pl(k,h) 


P  lh,/c) 


Fig.  58. — Triangles  of  reference  for  complementary  angles. 


65.  Functions  of  Complementary  Angles.  Angles  are  said 
to  be  complementary  if  their  sum  is  90°.  Angles  are  said  to  be 
supplementary  if  their  sum  is  180°. 

Let  0  be  an  angle  in  the  first  quadrant,  and  draw  the  angle 
(90° -0)  of  terminal  side  OPi,  as  shown  in  Fig.  58.  Let  P  and  Pi 
lie  on  a  circle  of  radius  a.  Let  the  coordinates  of  the  point  P  be 
{h,  k),  then  Pi  is  the  point  {k,  h).  Hence  PiDi/OPi  =  h/a  = 
sin  (90° -5).    But  from  the  triangle  PDO,    h/a  =  cos  8.    Hence 


Likewise, 


sin  (90°  —  6)  =  cos  6 
tan  (90°  -  0)  =  cot  e 
sec  (90°  -  d)  =  esc  e 


These  relations  explain  the  meaning  of  the  words  cosine,  co- 
tangent, cosecant,  which  are  merely  abbreviations  for  comple- 


§56]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    115 


merit's  sine,  complement's  tangent,  etc.  Collectively,  cosine, 
cotangent,  and  cosecant  are  called  the  co-functions.  Likewise, 
from  Fig.  58, 

cos  (90°  —  6)  =  siad 

cot  (90°  -  0)  =  tan  e 

CSC  (90°  —  d)  =  sec  d 

Later  it  will  be  shown  that  the  above  relations  hold  for  all 
values  of  d,  positive  or  negative. 

56.  Graph  of  the  Sine  and  Cosine.  In  rectangular  coordinates 
we  can  think  of  the  ordinate  y  of  a  point  as  depending  for  its 
value  upon  the  abscissa  or  x  of  that  point  by  means  of  the  equation 
y  =  sin  X,  provided  we  think  of  each  value  of  the  abscissa  laid 
off  on  the  Z-axis  as  standing  for  some  amount  of  angular  mag- 
nitude.    Therefore  the  equation  y  =  sinx  must  possess  a  graph 


Y 

A 

e 

C 

p 

^ 

- 

- 

7 

^ 

- 

- 

i. 

-- 

-- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

A 

/ 

s 

^   \\ 

A 

/ 

« 

-TT 

A 
/ 

1/ 

s 

p 

0 

s. 

/v 

B     D,D/^ 

1 

s 

/ 

A 

s 

/ 

y 

/ 

\ 

/ 

y 

/ 

\ 

/ 

y  y 

^_ 

- 

- 

- 

- 

— 

^ 

- 

- 

- 

- 

^ 

- 

-- 

-q 

|h 

- 

- 

- 

- 

V 

s 

- 

- 

- 

- 

- 

-^ 

2 

^ 

- 

- 

= 

X 

L. 

_ 

_ 

_ 

„ 

_ 

_ 

_ 

_ 

_ 

a 

5 

^^ 

2-1 

r- 

_ 

_ 

„ 

^ 

_ 

_ 

^ 

^ 

T- 

^ 

_ 

_ 

^ 



-> 

Fig.  59. — Construction  of  the  sinusoid. 

in  rectangular  coordinates.  In  order  to  produce  the  graph  of 
y  =  sin  X,  it  is  best  to  lay  off  the  angular  measure  x  on  the  X- 
axis  in  such  a  manner  that  it  may  conveniently  be  thought  of 
in  either  radian  or  degree  measure.  If  we  suppose  that  a  scale 
of  inches  and  tenths  is  in  the  hands  of  the  student  and  that  a 
graph  is  required  upon  an  ordinary  sheet  of  unruled  paper  of 
letter  size  (8|-  X  11  inches),  then  it  will  be  convenient  to  let 
1/5  inch  of  the  horizontal  scale  of  the  X-axis  correspond  to  10° 
or  to  ir/18  radians  of  angular  measure.  To  accomplish  this, 
the  length  of  one  radian  must  be  1.15  inches  (i.e.,  18/5t  inch), 
which  length  must  be  used  for  the  radius  of  the  circle  on  which 
the  arcs  of  the  angles  are  laid  off.    Hence,  to  graph  y  =  sin  x, 


116        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§56 

draw  at  the  left  of  a  sheet  of  (unruled)  drawing  paper  a  circle  of 
rddius  1.15  inches,  as  the  circle  OP5B,  Fig.  59.  Take  0  as  the 
origin  and  prolong  the  radius  BO  for  the  positive  portion  OX 
of  the  X-axis.  Divide  OL  into  1/5-uich  intervals,  each  corre- 
sponding to  10°  of  angle;  eighteen  of  these  correspond  to  the 
length  IT,  if  the  radius  BO  (1.15  inches)  be  the  unit  of  measure. 
Next  divide  the  F-axis  proportionately  to  sin  x  in  the  following 
manner :  With  a  pair  of  bow  dividers,  or  by  means  of  a  protractor, 
divide  the  semicircle  into  eighteen  equal  divisions  as  shown 
in  the  figure,  thus  making  the  length  of  each  small  arc  exactly 
1/5  inch.  The  perpendiculars,  or  ordinates,  dropped  upon  OX 
from  each  point  of  division,  divided  by  the  radius,  are  the  sines 
of  the  corresponding  angles.  Draw  lines  parallel  to  OX  through 
each  point  of  division  of  this  circle. '  These  cut  the  F-axis  at  points 
Ai,  Ai,  .     Then  if  the  radius  of  the  circle  be  called  unity, 

the  distances  OAi,  OA 2,  OA3,  .  .  are  respectively  the  sines  of 
the  angles  OBPi,  OBP2,  OBP3,  These  are  the  successive 

ordinates  corresponding  to  the  abscissas  already  laid  off  on  OL. 
The  curve  is  then  constructed  as  follows :  First  draw  vertical  lines 
through  the  points  of  division  of  OX;  these,  with  the  horizontal 
lines  already  drawn,  divide  the  plane  into  a  large  number  of  rec- 
tangles. Starting  at  0  and  sketching  the  diagonals  (curved  to 
fit  the  alignment  of  the  points)  of  successive  "cornering"  rec- 
tangles, the  curve  OCNTL  is  approximated,  which  is  the  graph 
oi  y  —  sin  x.  This  curve  is  called  the  sinusoid  or  sine  curve. 
The  curve  is  of  very  great  importance  for  it  is  found  to  be  the 
type  form  of  the  fundamental  waves  of  science,  such  as  sound 
waves,  vibrations  of  wires,  rods,  plates  and  bridge  members, 
tidal  waves  in  the  ocean,  and  ripples  on  a  water  surface.  The 
ordinary  progressive  waves  of  the  sea  are,  however,  not  of  this 
shape.  Using  terms  borrowed  from  the  language  of  waves,  we 
may  call  C  the  crest,  TV  the  node,  and  T  the  trough  of  the  sinusoid. 
It  is  obvious  that  as  x  increases  beyond  27r'',  the  curve  is  re- 
peated, and  that  the  pattern  OCNTL  is  repeated  again  and  again 
both  to  the  left  and  to  the  right  of  the  diagram  as  drawn.  Thus 
it  is  seen  that  the  sine  is  a  periodic  function  of  period  2t'  or  360°. 

\For  lack  of  room  only  a  few  of  the  successive  points  Pi,  P2,  P3,  ,    ,  ,  of  (iivigign 
ef  the  quadrant  OPjPf  are  actually  lettered  in  Fig.  59, 


§67]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   117 

The  small  rectangles  lying  along  the  X-axis  are  nearly  squares. 
They  would  be  exactly  equilateral  if  the  straight  Hne  OAi  was  equal 
to  the  arc  OPi.  This  equality  is  approached  as  near  as  we  please 
as  the  number  of  corresponding  divisions  of  the  circle  and  of  OX 
is  indefinitely  increased.  In  this  way  we  arrive  at  the  notion  of 
the  slope  of  a  curve  in  mathematics.  In  this  case  wfe  say  that  the 
slope  of  the  sinusoid  at  0  is  +  1  and  at  A'^  is  —  1,  and  at  L  is  +  1. 
We  say  that  the  curve  outs  the  axis  at  an  angle  of  45°  at  0  and 
at  an  angle  of  315°  (or  —  45°  if  we  prefer)  at  N.  The  slope  at  C 
and  at  T  is  zero. 

The,  curve  y  =  a  sin  x  is  made  from  y  =  sin  x  by  multiplying 
all  of  the  ordinates  of  the  latter  by  a.  The  number  a  is  called 
the  amplitude  of  the  sinusoid. 

57.  The  Cosine  Curve.  In  Fig.  60  let  the  angles  COPi,  COP2, 
COP3,  etc.,  be  laid  off  from  the  position  of  the  F-axis  OY  as  initial 
side.    Then  if  the  radius  of  the  circle  be  called  unity,  the  dis- 


Y 

^Z), 

0 

D 
D 

¥- 

s; 

f^^  \  \ 

R/\  \  \  \ 

N 

' 

^ 

1 

/\  \\\\ 

D. 

S 

/ 

/      ^\\\\ 

s 

/ 

/    ^>A\\\\ 

s 

/ 

\ 

A 

L 

/ 

A, 

7r 

\ 

or 

/ 

H 

TT 

27r 

? 

S 

/ 

? 

\ 

\ 

/ 

\ 

s 

/ 

/ 

_ 

_ 

_ 

_ 

— 

— 

— 

^ 

— 

— 

— 

^ 

— 

_ 

_ 

— 

— 

T 

Y 

1 

t- 

Fig.  60. — Construction  of  the  cosine  curve. 


tances  ODj,  OD2,  OD3,  .    .       are  respectively  the  cosines  of  the 
angles  COP\,  COP^,  COP3,  .     If  the  distances  laid  off  on  the 

Z-axis  represent  the  measures  of  the  successive  angles  COPi,  COP2, 
then  the  curve  shown  in  the  figure  has  the  equation 
y  =  cos  X.  The  construction  shows  that  the  curve  is  exactly  the 
same  as  the  sine  curve  of  Fig.  59  except  that  the  origin  for  the 
cosine  curve  is  under  the  crest  while  in  the  sine  curve  the  origin 
is  at  a  node.  If  the  origin  be  taken  at  0'  in  Fig.  59  the  curve  may 
be  called  the  cosine  curve. 


118       ELEMENTAEY  MATHEMATICAL  ANALYSIS       [§68 

In  Fig.  62  the  curve  ABODE  is  the  cosine  curve  y  =  cos  x.    The 
other  curve  is  the  sine  curve  y  =  sin  x. 

68.  The  Sine  of  a  Negative  Angle.    In  Fig.  61  the  full  drawn 
curve  represents  the  graph  for  y  =  sin  x.    The  graph  for 

y  =  sin  (—  x)  (1) 

r 


FiQ.  61. — The  relation  between  y  =  aia  x  and  y  =  sin  (—  s). 


may  be  obtained  by  rotating  the  graph  for  y  =  sin  x, 
180°  about  the  F-axis,  by  Theorem  I  on  Loci.  This  gives  the 
dotted  curve  of  Fig.  61.  But  from  the  properties  of  the  sinusoid, 
the  dotted  curve  is  the  reflection  in  the  Z-axis  of  the  curve  drawn 
in  full,  hence  the  equation  of  the  dotted  curve  may  also  be  written 

— 2/  =  sin  X.  (2) 

Hence,  from  (1)  and  (2) 

sin  (— x)  =  —sin  x.  (3) 

69.  Complementary   Angles.    Fig.   62  shows  the   curves  for 
y  =  cos  X  and  for  y  =  sin  x.    By  the  properties  of  these  curves 


Fig.  62. — Comparison  of  the  sine  and  cosine  curves. 

it  is  obvious  that  the  cosine  curve  may  be  regarded  as  the  sine 
curve  translated  x/2  units  to  the  left.    That  is,  the  cosine  curve 

y  =  cos  X  (1) 

has  also  the  equation 

y  =  sin  (a;  +  I)  •  (2) 


§60]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   119 

Since  this  curve  (the  cosine  curve)  is  symmetrical  about  the  Y- 
axis,  its  equation  remains  unchanged  if  we  change  x  to  {—x), 
by  Theorem  I  on  Loci.  Hence  the  cosine  curve  has  also  the 
equation 

y  =  sin  (  -  ^  +  |\  =  smi^-x\-  (3) 

Comparing  (1)  and  (3)  we  see  that  we  have  proved  for  all  values 
of  X  that 

sin  ( x|  =  cosx.  (4) 

By  comparing  (1)  with  (2)  we  see  that  sin  (s  +  *)    ~  ^°^  ^' 

this  fact  is,  however,  niuch  less  useful  than  that  represented  by 
equation  (4). 

Exercises 

From 'the  curves  for  y  =  sin  x  and  y  =  sin(  — x),  Fig.  61  shows 
that: 

1.  sin  {x  —  v)  =  sin  {—x)  and  hence  sin  (tt  —  s)  =  sin  x. 
From  the  curves  for  y  =  cos  x  and  y  =^  sxixx,  Fig.  62,  shows  that: 

2.  sin  X  =  cos  {x  —  s).  3.  cos  x  =  sin  {x  —  fir). 
4.  cos  {x  +  -fir)  =  sin  x.  5.  cos  (.—x)  =  cos  x. 

60.  Trigonometric  Functions  of  Negative  Angles.  We  have 
already  shown,  (3)  §57,  that 

sin  (—  x)  =  —  sin  x.  (1) 

Also  from  Fig.  62,  since  the  cosine  curve  is  symmetrical  about  the 
y-axis, 

cos  (— x)  =  cos  X.  (2) 

Dividing  the  members  of  (1)  by  the  members  of  (2)  we  find 

tan  (—  x)  =  tan  x.  (3) 

61.  Odd  and  Even  Fimctions.  A  function  that  changes  sign 
but  retains  the  same  numerical  value  when  the  sign  of  the  argu- 
ment is  changed  is  called  an  odd  function.  Thus  sin  x  is  an  odd 
function  of  x,  since  sin  (—x)  =  —sin  x.  Likewise  x^  is  an  odd 
function  of  x,  as  are  all  odd  powers  of  x.  The  graph  of  an  odd 
function  of  a;  is  symmetrical  with  respect  to  the  origin  0 ;  that  is, 


120        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§62 

if  P  is  a  point  on  the  curve,  then  if  the  line  OP  be  produced 
backward  through  0  a  distance  equal  to  OP  to  a  point  P',  then 
P'  also  lies  on  the  curve.  The  parts  of  2/  =  x'  in  the  first 
and  third  quadrants  are  good  illustrations  of  this  property. 

A  function  of  x  that  remains  unaltered,  both  in   sign  and 
numerical  value,  when  the  argument  is  changed  in  sign,  is  called 
an  even  function  of  x.    Examples  are  cos  x,  x',  x^  —  3x*, 
The  graph  of  an  even  function  is  symmetrical  with  respect  to  the 
y-axis. 

Most  functions  are  neither  odd  nor  even,  but  mixed,  like 
x^  +  sin  X,  x^  -\-  x',  and  x  +  cos  x. 

Exercises 

1.  Is  sin's  an  odd  or  an  even  function  of  x1  Is  tan'a;  an  odd  or 
an  even  function  of  x7 

2.  Is  the  function  sin  x  +  2  tan  x  an  odd  or  an  even'functfon?  Is 
sin  X  +  cos  x  an  odd  or  an  even  function  of  a;? 

62.  The  Defining  Equations  Cleared  of  Fractions.  The  student 
should  commit  to  memory  the  equations  defining  the  trigonometric 
functions  when  cleared  of  fractions.  In  this  form  the  equations 
are  quite  as  useful  as  the  original  ratios.     They  are  written: 

y  =  a  sin  6  y  =  x  tan  6  a  =  x  sec  6 

X  =  a  cos  6  X  =  J  cot  d  a  =  y  esc  9 

As  applied  to  the  right  angled  triangle,  these  three  sets  of  equa- 
tions may  be  stated  in  words  as  follows: 

Either  leg  of  a  right  triangle  is  equal  to  the  hypotenuse  multiplied 
by  the  sine  of  the  opposite,  or  by  the  cosine  of  the  adjacent,  angle. 

Either  leg  of  aright  triangle  is  equal  to  the  other  leg  multiplied  by  the 
tangent  of  the  opposite,  or  by  the  cotangent  of  the  adjacent,  angle. 

The  hypotenuse  of  a  right  triangle  is  equal  to  either  leg  multiplied 
by  the  secant  of  the  angle  adjacent,  or  by  the  cosecant  of  the  angle 
opposite  that  leg. 

These  statements  should  be  committed  to  memory. 

63.  Orthographic  Projection.  In  elementary  geometry  we 
learned  that  the  projection  of  a  given  point  P  upon  a  given  line  or 
plane  is  the  foot  of  the  perpendicular  dropped  from  the  given  point 


§63]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    121 

upon  the  given  line  or  plane.  Likewise  if  perpendiculars  be 
dropped  from  the  end  points  A  and  B  of  any  line  segment  AB  upon 
a  given  line  or  plane,  and  if  the  feet  of  these  perpendiculars  be 
called  P  and  Q,  respectively,  then  the  line  segment  PQ  is  called  the 
projection  of  the  line  AB.  Also,  if  perpendiculars  be  dropped 
from  all  points  of  a  given  curve  AB  upon  a  given  plane  MO,  the 
locus  formed  by  the  feet  of  all  perpendiculars  so  drawn  is  called  the 
projection  of  the  given  curve  upon  the  plane  MO. 

To  emphasize  the  fact  that  the  projections  were  made  by. using 
perpendiculars  to  the  given  plane,  it  is  customary  to  speak  of  them 
as  orthogonal  or  orthographic  projections. 


Pig.  63. — Orthographic  projection  of  line  segments. 


The  shadow  of  a  hoop  upon  a  plane  surface  is  not  the  ortho- 
graphic projection  of  the  hoop  unless  the  rays  of  light  from  the  sun 
strike  perpendicular  to  the  surface.  This  could  only  happen  in 
our  latitude  upon  a  suitable  non-horizontal  surface. 

The  shortening,  by  a  given  fractional  amount,  of  a  set  of 
parallel  line  segments  of  a  plane  may  be  brought  about  geometric- 
ally by  orthographic  projection  of  all  points  of  the  line  segments 
upon  a  second  plane.  For,  in  Fig.  63,  let  AiBi,  A^Bi,  A3B3, 
etc.,  be  parallel  line  segments  lying  in  the  plane  MN.  Let  their 
projections  on  any  other  plane  be  Aid',  AiC^',  Ai'C-/,  etc., 
respectively.     Draw  A\.C\  parallel  to  Ai'Ci'  and  Aid  parallel  to 


122        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§63 

Aj'Cj',  etc.    Then  since  the  right  triangles  AiBiCi,  AiB^Ct, 
AiBaCz,  etc.  are  similar, 

AiBi      A2B2      AsBs 


AiCi      A2C2      AiC 

Call  this  ratio  a.  It  is  evident  that  a>l.  Substitute  the 
equals:  A/d'  =  AiCi,  Aj'Ca'  =  A^d,  etc.    i'hen 

AiBi  _  A2B2  _  A3B3  _  ,  _a 

Ai'Ci' ~ Ai'Ci' " As'c ~      '  ~r 

The  numerators  are  the  original  lii\e  segments;  the  denominators 
are  their  projections  on  the  plane  MO.  The  equality  of  these 
fractions  shows  that  the  parallel  lines  have  all  been  shortened  in 
the  ratio  a:  I. 

The  above  work  shows  that  to  produce  the  curve  y  =  (x/a)", 
(o  <  1),  from  2/  =  a;"  by  orthographic  projection  it  is  merely  neces- 
sary to  project  all  of  the  abscissas  oi  y  =  x"  upon  a  plane  passing 
through  YOY'  making  an  angle  with  OX  such  that  unity  on  OX 
projects  into  a  length  a  on  the  projection  of  OX.  To  produce  the 
curve  y  =  ax"  (a  <  1)  from  y  =  x"  hy  orthographic  projection  it  is 
merely  necessary  to  project  all  of  the  ordinates  oi  y  =  x"  upon  a 
plane  passing  through  XOX'  making  an  angle  with  OY  such  that 
unity  on  OY  projects  into  the  length  a  on  the  projection  of  OY. 

To  lengthen  all  ordinates  of  a  given  curve  in  a  given  ratio,  1 :  a,  the 
process  must  be  reversed;  that  is,  erect  perpendiculars  to  the  plane 
of  the  given  curve  at  all  points  of  the  curve,  and  cut  them  by  a^lane 
passing  through  XOX'  making  an  angle  with  OY  such  that  a  length 
a  (,a>  1)  measured  on  the  new  K-axis  projects  into  unity  on  OF  of 
the  original  plane. 

In  Fig.  50  the  projection  of  OP  in  any  of  its  positions,  such  as 
OPi,  OP2,  OP3,  ■  ■ .,  is  ODi,  OD2,  OD3,  . . . ,  or  is  the  abscissa  of 
the  point  P.    Thus  for  all  positions 

X  =  a  cos  6. 

The  sign  of  x  gives  the  sign,  or  sense,  of  the  projection.  In  each 
case  0  is  said  to  be  the  angle  of  projection. 

This  definition  of  projection  is  more  general  in  one  respect  than 
that  discussed  above.  By  the  present  definition  the  projection  of 
a  line  is  negative  if  90°  <  9  <  270°  (read,  "if  6  is  greater  than  90° 


§64]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    123 

but  is  less  than  270°").  This  concept  is  important  and  essential 
in  expressing  a  component  of  a  displacement,  of  a  velocity,  of  an 
acceleration,  or  of  a  force. 

The  cosine  of  6  might  have  been  defined  as  that  proper  fraction 
by  which  it  is  necessary  to  multiply  the  length  of  a  line  in  order  to 
produce  its  projection  on  a  line  making  an  angle  d  with  it. 

Exercises 

1.  A  stretched  guy  rope  75  ft.  long  makes  an  angle  of  60°  with  the 
horizontal.  What  is  the  length  of  the  projection  of  the  rope  on  a 
horizontal  plane?  What  is  the  length  of  the  projection  of  the  rope 
on  a  vertical  plane? 

2.  Find  the  lengths  of  the  projections  of  the  line  through  the  origin 
and  the  point  (1,  -y/s)  upon  the  OX  and  OF  axes,  if  the  Une  is  12  inches 
long. 

3.  A  line  8  inches  long  makes  an  angle  of  45°  with  the  X-axis. 
What  is  the  length  of  its  projection  on  the  X-axis? 

4.  A  velocity  of  20  feet  per  second  is  represented  as  the  diagonal 
of  a  rectangle  the  longer  side  of  which  makes  an  angle  of  30°  with  the 
diagonal.  Find  the  components  of  the  velocity  along  each  side  of  the 
rectangle. 

5.  Show  that  the  projections  of  a  fixed  hne  OA  upon  all  other 
lines  drawn  through  the  point  0  are  chords  of  a  circle  of  diameter  OA . 
See  Fig.  66. 

6.  Find  the  projection  of  the  side  of  a  regular  hexagon  upon  the 
three  diagonals  passing  through  one  end  of  the  given  side,  if  the  side 
of  the  hexagon  is  20  feet. 

64.  Polar  Coordinates.  In  Fig.  64,  the  position  of  the  point 
P  may  be  assigned  either  by  giving  the  x  and  y  of  the  rectangular 
coordinate  system,  or  by  giving  the  vectorial  angle  6  and  the 
distance  OP  measured  along  the  terminal  side  of  6.  Unlike 
the  distance  o  used  in  the  preceding  work,  it  is  found  conven- 
ient to  give  the  line  OP  a  sense  or  direction  as  well  as  length; 
such  a  line  is  called  a  vector.  In  the  present  Case,  OP  is  known  as 
the  radius  vector  of  the  point  P,  and  it  is  usually  symbolized  by 
the  letter  p.  The  vectorial  angle  6  and  the  radius  vector  p  are 
together  called  the  polar  coordinates  of  the  point  P,  and  the 
system  used  in  locating  the  point  is  known  as  the  system  of  polar 


124        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§65 


co5rdinates.  In  Fig.  64  the  point  P'  is  located  by  turning  from 
the  fundamental  direction  OX,  called  the  polar  axis,  through  an 
angle  6  and  then  stepping  backward  the  distance  p  to  the  point 
P';  this  is,  then,  the  point  (—  p,  9).  P'  has  also  the  coordinates 
(p,  02),  in  which  6^  =  0  +  180°;  likewise  Pi  is  (+  p',  di)  and 
P'l  is  (—  p',  Bi).  Thus  each  point  may  be  located  in  the  polar 
system  of  coordinates  in  two  ways,  i.e.,  with  either  a  positive  or  a 
negative  radius  vector.  If  negative  values  of  B  be  used,  there 
are  four  ways  of  locating  a  point  without  using  values  oi  B> 
360°.    In  giving  a  point  in  polar  coordinates,  it  is  usual  to  name 

the  radius  vector  first  and  then 
the  vectorial  angle;  thus  (5, 40°) 
means  the  point  of  radius  vec- 
tor 5  and  vectorial  angle  40°. 

65.  Polar  Coordinate  Paper. 

Polar  coordinate  paper  (form 
MZ)  is  prepared  for  the  con- 
struction of  loci  in  the  polar 
system.  A  reduced  copy  of  a 
sheet  of  such  paper  is  shown 
in  Fig.  65.  This  plate  is  grad- 
uated in  degrees,  but  a  scale  of 
radian  measure  is  given  in  the 
margin.  The  radii  proceeding 
from  the  pole  0  meet  the  circles  at  right  angles,  just  as  the  two 
systems  of  straight  lines  meet  each  other  at  right  angles  in  rect- 
angular coordinate  paper.  For  this  reason,  both  the  rectangular 
and  the  polar  systems  are  called  orthogonal  systems  of  coordinates. 

We  have  learned  that  the  fundamental  notion  of  a  function  implies 
a  table  of  corresponding  values  for  two  variables,  one  called  the 
argument  and  the  other  the  function.  The  notion  of  a  graph  implies 
any  sort  of  a  scheme  for  a  pictorial  representation  of  this  table  of 
values.  There  are  three  common  methods  in  use:  the  double  scale, 
the  rectangular  coordinate  paper,  and  the  polar  paper.  The  polar 
paper  is  very  convenient  in  case  the  argument  is  an  angle  measured 
in  degrees  or  in  radians.  Since  in  a  table  of  values  for  a  functional 
relation  we  need  to  consider  both  positive  and  negative  values  for 
both  the  argument  and  the  function,  it  is  necessary  to  use  on  the 


Polar  coordinates. 


§65]     THE  CIRCLE  AND  THE  CIRCULAR;  FUNCTIONS    125 

polar  paper  the  convention  already  explained.  The  argument,  which 
is  the  angle,  is  measured  counter-clockwise  if  positive  and  clockwise 
if  negative  from  the  line  numbered  0°,  Fig.  65.  The  function  is 
measured  outward  from  the  center  along  the  terminal  side  of  the  angle 
for  positive  functional  values  and  outward  from  the  center  along  the 
terminal  side  of  the  angle  produced  backward  through  the  center  for 
negative  functional  values.  In  this  scheme  it  appears  that  four  differ- 
ent pairs  of  values  are  represented  by  the  same  point.     This  is  made 


FiQ.  65. — Polar  coordinate  paper. 


clear  by  the  points  plotted  in  the  figure.  The  points  Pi,  Pi,  Pa,  Pt 
are  as  follows: 

Pi:  (6,  40°);  (6,  -  320°);  (-  6,  220°);  (-  6,  -  140°). 

Ps:  (10,  135°);  (10,  -  225°);  (-  10,  315°);  (-10,  -  45°). 

Pa:  (5,  230°);  (5,  -  130°);  (-  5,  50°);  (-  5,  -  310°). 

Pi-.  {,&,  330°);  (6,  -  30°);  (-  6,  150°);  (-  6,  -  210°). 

The  angular  scale  cannot  be  changed,  but  the  functional  scale 
can  be  changed  at  pleasure. 

In  case  the  vectorial  angle  is  given  in  radians,  the  point  may 


126        ELEMENTARY  MATHEMATICAL  ANALYSIS 

be  located  on  polar  paper  by  means  of  a  straight  edge  and  the 
marginal  scale  on  form  M3. 

The  point  0,  Fig.  65,  is  called  the  pole  and  the  line  OA,  the 
polar  axis. 

Exercises 

1.  Plot  upon  polar  coordinate  paper  the  following:  (a)  (0.1,  30°;; 
(6)  (0.2,40°);  (c)  (0.6,120°);  (d)  (0.8,-30°);  (e)  (1.2,300°); 
(/)  (0.7,  -  47°).    Let  10  cm.  =  1  unit  for  p. 

2.  Plot  upon  polar  coordinate  paper  the  following:  (a)  (1.3,  45°); 
(6)  (11.1,  137°);  (cj  (9.2,  -  47°);  (d)  (8.5,  -  216).  Let  1  cm.  =  1  unit 
for  p. 

3.  Plot  upon  polar  coordinate  paper  the  following:  (a)  (10,  C); 
(6)  (9,  D;  (c)  (8.2,  1.6');  (d)  (12,  3.2"-).     Let  1  cm.  =  1  unit  for  p. 

4.  Explain  why  the  locus  for  p  =  3  is  a  circle  with  center  at  the 
pole  and  radius  equal  to  three  units. 

5.  Draw  the  loci  for  p  =  5  and  p  =  7. 

6.  Explain  why  the  locus  8  =  J  tt  is  a  straight  line  passing  through 
the  pole  and  making  an  angle  of  45°  with  the  polar  axis.  Explain  why 
this  locus  is  indefinite  in  extent  and  does  not  terminate  at  the  pole. 

7.  Draw  loci  for:  9  =  |  tt,  and  d  =  —  \ir. 

8.  Plot  the  locus  for  p  =  9,  if  9  is  measured  in  radians.  Use  2  cm. 
as  the  unit  for  p. 

66.  Graphs  of  p  =  a  cos  9  and  p  =  a  sin  0.  These  are  two  funda- 
mental graphs  in  polar  coordinates.  The  equation  p  =  o  cos  6 
states  that  p  is  the  projection  of  the  fixed  length  a  upon  a  radial 
line  proceeding  from  0  and  making  a  direction  angle  6  with  a, 
or,  in  other  words,  p  in  all  of  its  positions  must  be  the  side  adjacent 
to  the  direction  angle  0  in  a  right  triangle  whose  hypotenuse  is  the 
given  length  o.  (See  §62  and  Fig.  66.)  It  must  be  remem- 
bered that  the  direction  angle  d  is  always  measured  from  the  fixed 
direction  OA.  Hence,  to  construct  the  locus  p  =  a  cos  6,  proceed 
as  follows:  Draw  a  number  of  radical  lines  from  0,  Fig.  66. 
Project  upon  each  of  these  the  constant  length  OA,  or  a.  These 
projections  are  then  radius  vectors  for  p  =  a  cos  0  and  a  curve 
drawn  through  their  end  points  gives  the  required  locus. 

Thi  locus  is  a  circle  since  P  is  always  at  the  vertex  of  a  right 
triangle  standing  on  the  fixgd  hypotenuse  a,  and  therefore  the 
point  P  is  on  the  semicircle  AOP;  for,  from  plane  geometry,  a  right 
triangle  is  always  inscribable  in  a  semicircle. 


§66]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   127 

When  6  is  in  the  second  quadrant,  as  62,  Fig.  66,  the  cosine  is 
negative  and  consequently  p  is  negative.  Therefore  the  point 
P2  is  located  by  measuring  backward  through  0.  Since,  however, 
P2  is  the  projection  of  a  through  the  angle  62  (see  §63),  the 
angle  at  P2  must  be  a  right  angle.  Thus  the  semicircle  OP2A 
is  described  as  d  sweeps  the  second  quadrant.  When  6  is  in 
the  third  quadrant,  as  ds,  the  cosine  is  still  negative  and  p  is 
measured  backward  to  describe  the  semicircle  APiO  a  second 
time.  As  6  sweeps  the  fourth  quadrant,  the  semicircle  OP2A  is 
described  the  second  time.     Thus  the  graph  in  polar  coordinates 


Fig.  66. — The  graph  of  p  =  a  cos  e. 


of  p  =  a  cos  d  is  a  circle  twice  drawn  as  6  varies  from  0°  to  360°. 
Once  around  the  circle  corresponds  to  the  portion  ABC  of  the 
"wave"  y  =  a  cos  x,  in  Fig.  61.  The  second  time  around  the 
circle  corresponds  to  the  portion  CDE  from  trough  to  crest  of  the 
cosine  curve.  Trough  and  crest  of  all  the  successive  "wave 
lengths"  correspond  to  the  point  A,  the  nodes  to  the  point  0. 

The  polar  representation  of  the  cosine  of  a  variable  bj^  means 
of  the  circle  is  more  useful  and  important  in  science  than  the 
Cartesian  representation  by  means  of  the  sinusoid.  The  ideas 
here  presented  should  be  thoroughly  mastered  by  the  student. 

The  graph  of  p  =  a  sin  6  is  also  a  circle,  but  the  diameter  is 
the  line  OB  making  an  angle  of  90°  with  OA,  as  shown  in  Fig.  67. 
Since  p  =  a  sin  6,  the  radius  vector  must  equal  the  side  lying 
opposite  the  angle  6  in  a  right  triangle  of  hypotenuse  a,   if 


128        ELEMENTARY  MATHEMATICAL  ANALYSIS        ['§67 

0°  <  e  <  90°.  Since  angle  AOPi  =  angle  OBPi,  the  point  Pi 
is  the  vertex  of  any  right  triangle  erected  on  OB,  or  a,  as  a  hypote- 
nuse. The  semicircle  BP2O  is  described  as  0  increases  from  90°  to 
180°.  Beyond  180°  the  sine  is  negative,  so  that  the  radius  vector 
p  must  be  laid  off  backward  for  such  angles.  Thus  P3  is  the 
point  corresponding  to  the  angle  63  of  the  third  quadrant.  As  6 
sweeps  the  third  and  fourth  quadrants  the  circle  OP1BP2O  is 
described  a  second  time.  Therefore  the  graph  of  p  =  asiad 
is  the  circle  tiince  drawn  of  diameter  a,  and  tangent  to  OA  at  0. 
The  first  time  around  the  circle  corresponds  to  the  crest,  the 
second  time  around  corresponds  to  the  trough  of  the  wave  or 
sinusoid  drawn  in  rectangular  coordi- 
nates. 0  corresponds  to  the  nodes  of 
the  sinusoid  and  B  to  the  maximum  and 
minimum  points,  or  to  the  crests  and 
troughs. 

We  have  seen  that  the  graph  of  a 
function  in  polar  coordinates  is  a  very 
different  curve  from  its  graph  in  rect- 
angular coordinates.  Thus  the  cosine 
of  a  variable  if  graphed  in  rectangular 
coordinates  is  a  sinusoid;  but  if  graphed 

FiQ.  g7_ xhe  graph  of  i^  polar  coordinates  it  is  a  circle  (twice 

p  =  a  sine.  drawn) .     There   is  in  this  case  a  very 

great  difference  in  the  ease  with  which 
these  curves  can  be  constructed;  the  sinusoid  requires  an  elabo- 
rate method,  while  the  circle  may  be  drawn  at  once  with  com- 
passes. This  is  one  reason  why  the  periodic,  or  sinusoidal  rela- 
tion, is  preferably  represented  in  the  natural  sciences  by  polar 
coordinates. 

Exercises 

1.  Show  that  if  —  o  is  negative,  p  =  —  a  cos  9  is  a  circle,  diameter 
a,  with  center  to  left  of  the  pole  §a  units. 

2.  Show  that  if  —  a  is  negative  p  —  —  a  sin  0  is  a  circle,  diameter 
o,  with  center  below  the  pole  50  units. 

67.  Graphical  Table  of  Sines  and  Cosines.  The  polar  graphs 
of  p  =  a  sin  0  and  p  =  a  cos  9  furnish  the  best  means  of  construct- 
ing graphical  tables  of  sines  and  cosines.     The  two  circles  passing 


§68]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   129 

through  0  shown  on  the  polar  coordinate  paper,  form  M3,  Fig.  65, 
are  drawn  for  this  purpose.  A  supply  of  this  coordinate  paper 
should  be  in  the  hands  of  the  student.  If  the  diameter  of  the 
sine  and  cosine  circles  be  called  1,  then  the  radius  vector  of  any 
point  on  the  lower  circle  is  the  cosine  of  the  vectorial  angle,  and 
the  radius  vector  of  the  corresponding  point  on  the  upper  circle 
,  is  the  sine  of  the  vectorial  angle.  Thus,  from  the  diagram  of 
form  M3,  we  read  cos  45°  =  0.707;  cos  60°  =  0.500;  cos  30°  = 
0.866.    These  results  are  correct  to  the  third  place. 

Exercises 

1.  /From  coordinate  paper,  form  M3,  find  the  values  of  the  following : 
(a)  cos  36°;  (b)  cos  62°;  (c)  cos  126°;  (,d)  sin  81°;  (e)  sin  25°;  (/)  sin  226°. 

68.  Graphical  Table  of  Tangents  and  Secants.  Referring  to 
Fig.  65,  it  is  obvious  that  the  numerical  values  of  the  tangents 
of  angles  can  be  read  off  by  use  of  the  uniform  scale  bordering 
the  polar  paper,  form  M3.  The  scale  referred  to  lies  just  inside 
of  the  scale  of  radian  measure,  and  is  numbered  0,  2,  4, 
Thus  to  get  the  numerical  value  of  tan  40°  it  is  merely  necessary  to 
call  unity  the  side  OA  of  the  triangle  of  reference  OAP,  and  then 
read  the  side  AP  =  0.84;  hence  tan  40°  =  0.84.  To  the  same 
scale  (i.e.,  OA  =  1)  the  distance  OP  =  1.31,  but  this  is  the  secant 
of  the  angle  AOP,  whence  sec  40°  =  1.31.  By  use  of  the  circles 
we  find  sin  40°  =  0.64  and  cos  40°  =  0.76. 

In  case  we  are  given  an  angle  greater  than  45°  (but  less  than 
135°)  use  the  horizontal  scale  through  B.  Starting  from  B  as 
zero  the  distance  measur/ed  on  the  horizontal  scale  is  the  cotangent 
of  the  given  angle.  The  tangent  is  found  by  taking  the  reciprocal 
of  the  cotangent. 

Exercises 

Find  the  unknown  sides  and  angles  in  the  following  right  triangles. 
The  numerical  values  of  the  trigonometric  functions  may  be  taken 
from  the  polar  paper.  The  vertices  of  the  triangles  are  supposed 
to  be  lettered  A,  B,  C  with  C  at  the  vertex  of  the  right  angle.  The 
small  letters  a,  b,  c  represent  the  sides  opposite  the  angles  of  the  same 
name.  See  also  table  of  Natural  Trigonometric  Functions  at  end  of 
the  book. 
9 


130        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§68 

By  angle  of  elevation  of  an  object  ia  meant  the  angle  between  a 
horizontal  line  and  a  line  to  the  object,  both  drawn  from  the  point  of 
observation,  when  the  object  lies  above  the  horizontal  line.  The  simi- 
lar angle  when  the  object  lies  below  the  observer  is  called  the  angle  of 
depression  of  the  object. 

The  solution  of  each  of  the  following  problems  must  be  cheeked. 
The  easiest  check  is  to  draw  the  triangles  accurately  to  scale  on  form 
Ml,  measuring  the  unknown  sides  and  angles. 

1.  When  the  altitude  of  the  sun  is  40°,  the  length  of  the  shadow  cast 
by  a  flag  pole  on  a  horizontal  plane  is  90  feet.  Find  the  height  of  the 
pole. 

Outline  of  Solution.  Call  height  of  pole  a,  and  length  of  shadow 
b.     Then  A  =  40°  and  B  =  50°.     Hence, 

o  =  6  tan  40°. 

Determining  the  numerical  value  of  the  tangent  from  the  polar  paper, 
we  find 

a  =  90  X  0.84  =  75.6  ft., 

which  result,  if  checked,  is  the  height  of  the  pole.  To  check,  either 
draw  a  figure  to  scale,  or  compute  the  hypotenuse  c,  thus : 

c  =  90  sec  40° 

From  the  polar  paper  find  sec  40°.     Then 

c  =  90  X  1.31  =  117.9 

Since  a^  +  b'  =  c',  we  have  c'  -  b^  =  a'',  or  (c  -  6)  (c  +  6)  =  a'. 
Hence,  if  the  result  found  be  correct, 

(117.9  -  90)  (117.9  +  90)  =  (75.6)  ^ 
5800  =  5715. 

These  results  show  that  the  work  is  correct  to  about  three  figures,  for 
the  sides  of  the  triangle  are  proportional  to  the  square  roots  of  the 
numbers  last  given. 

2.  At  a  point  200  feet  from,  and  on  a  level  with,  the  base  of  a  tower 
the  angle  of  elevation  of  the  top  of  the  tower  is  observed  to  be  60°. 
What  is  the  height  of  the  tower? 

3.  A  ladder  40  feet  long  stands  against  a  building  with  the  foot  of 
the  ladder  15  feet  from  the  base  of  the  wall.  How  high  does  the  ladder 
reach  on  the  wall? 

4.  From  the  top  of  a  vertical  cliff  the  angle  of  depression  of  a  point 


§68]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    13.1 

on  the  shore  150  feet  from  the  base  of  the  cliff  is  observed  to  be  30°. 
Find  the  heiglit  of  the  cliff. 

6.  In  walking  halt  a  mile  up  a  hill,  a  man  rises  300  feet.  Find  the 
angle  at  which  the  hill  slopes. 

If  the  hill  does  not  slope  uniformly  the  result  is  the  average  slope 
of  the  hill. 

6.  A  line  3.5  inches  long  makes  an  angle  of  35°  with  OX.  Find  the 
lengths  of  its  projections  upon  both  OX  and  OY. 

7.  A  vertical  cliff  is  425  feet  high.  From  the  top  of  the  cliff  the 
angle  of  depression  of  a  boat  at  sea  is  16°.  How  far  is  the  boat  from 
the  foot  of  the  chff? 

8.  The  projection  of  a  line  on  OX  is  7.5  inches,  and  its  projection 
on  OY  is  1.25  inches.  Find  the  length  of  the  line,  and  the  angle  it 
makes  with  OX. 

9.-  A  battery  is  placed  on  a  cliff  510  feet  high.  The  angle  of  depres- 
sion of  a  floating  target  at  sea  is  9°.  Find  the  range,  or  the  horizontal 
distance  of  the  target  from  the  battery. 

10.  From  a  point  A  the  angle  of  elevation  of  the  top  of  a  monument 
is  25°.  From  the  point  B,  110  feet  farther  away  from  the  base  of  the 
monument  and  at  the  same  elevation  as  A,  the  angle  of  elevation  is 
15°.     Find  the  height  of  the  monument  above  the  line  AB. 

11.  Find  the  length  of  a  side  of  a  regular  pentagon  inscribed  in  a 
circle  whose  radius  is  12  feet. 

12.  Proceeding  south  on  a  north  and  south  road,  the  direction  of  a 
church  tower,  as  seen  from  a  milestone,  is  41°  west  of  south.  From 
the  next  milestone  the  tower  is  seen  at  an  angle  of  65°  W.  of  S.  Find 
the  shortest  distance  of  the  tower  from  the  road. 

13.  A  traveler's  rule  for  determining  the  distance  one  can  see  from 
a  given  height  above  a  level  surface  (such  as  a  plain  or  the  sea)  is  as 
follows :  "  To  the  height  in  feet  add  half  the  height  and  take  the  square 
root.  The  result  is  the  distance  you  can  see  in  miles."  Show  that 
this  rule  is  approximately  correct,  assuming  the  earth  a  sphere  of 
raldius  3960  miles.  Show  that  the  drop  in  1  mile  is  8  inches,  and  that 
the  water  in  the  middle  of  a  lake  8  miles  in  width  stands  lOf  feet 
higher  than  the  water  at  the  shores. 

14.  Observations  of  the  height  of  a  mountain  were  taken  at  A  and 
B  on  the  same  horizontal  line,  and  in  the  same  vertical  plane  with  the 
top  of  the  mountain.  The  elevation  of  the  top  at  A  is  52°  and  at  B  is 
36°.     The  distance  AB  is  3500  feet.     Find  the  height  of  the  mountain. 

16.  The  diagonals  of  a  rhombus  are  16  and  20  feet.  Find  the 
lengths  of  the  sides  and  the  angles  of  the  rhombus. 


132        ELEMENTARY  MATHEMATICAL  ANALYSIS 


16.  The  equation  of  a  line  is  y  =  f.r  +10.  Compute  the  shortest 
distance  of  this  Une  from  the  origin. 

17.  Find  the  perimeter  and  area  of  ABCD,  Fig.  68. 

18.  Find  BC  and  the  total  area  of  ABCD,  Fig.  69. 

69.  The  Law  of  the  Circular  Functions.  It  will  be  emphasised 
in  this  book  that  the  fundamental  laws  of  exact  science  are  three  in 
number,  namely:  (1)  The  power  function  expressed  hy  y  =  ax" 
where  n  may  be  either  positive  or  negative;  (2)  the  harmonic  or 
periodic  law  y  =  aaia  nx,  which  is  fundamental  to  all  periodically 
occurring  phenomena;  and  (3)  a  law  to  be  discussed  in  a  sub- 
sequent chapter.  While  other  important  laws  and  functions 
arise  in  the  exact  sciences,  they  are  secondary  to  those  expressed 
by  the  three'  fundamental  relations. 


Fig.  68. — Diagram  for 
Exercise  17. 


Fig.  69. — Diagram  for 
Exercise  18. 


We  have  stated  the  law  of  the  power  function  in  the  following 
words  (see  §34): 

In  any  power  function,  if  x  change  hy  a  fixed  multipk,  y 
changes  by  a  fixed  multiple  also.  In  other  words,  if  x  change  by 
a  constant  factor,  y  will  change  by  a  constant  factor  also. 

Confining  our  attention  to  the  fundamental  functions,  sine 
and  cosine,  in  terms  of  which  the  other  circular  functions  can 
be  expressed,  we  may  state  their  law  as  follows  :i 


1  Chapter  XI  is  devoted  to  a  diacuasion  of  theae  fundamental  periodic  laws. 


§70]     THE  CIRCLE  AND  THE  CIRCULAK  FUNCTIONS   133 

The  circular  functions,  sin  6  and  cos  B,  change  periodically  in 
value  proportionally  to  the  periodic  change  in  the  ordinate  and 
abscissa,  respectively,  of  a  point  moving  uniformly  on  the  circle 
a;2  +  2/2  =  aK 

The  use  of  the  periodic  law  in  the  natural  sciences  is,  of  course, 
very  different  from  that  of  the  power  function.  The  student  will 
find  that  circular  functions  similar  toy  =  a  sin  nx  will  be  required 
in  order  to  express  properly  all  phenomena  which  are  recurrent 
or  periodic  in  character,  such  as  the  motion  of  vibrating  bodies, 
all  forms  of  wave  motion,  such  as  sound  waves,  light  waves, 
electric  waves,  alternating  currents  and  waves  on  water  surfaces, 
etc.  Almost  every  part  of  a  machine,  no  matter  how  compli- 
cated its  motions,  repeats  its  original  motions  at  stated  intervals 
and  these  recurrent  positions  are  expressible  in  terms  of  the 
circular  functions  and  not  otherwise.  The  student  will  obtain 
a  very  limited  and  unprofitable  idea  of  the  use  of  the  circular 
functions  if  he  deems  that  their  principal  use  is  in  numerical 
work  in  solving  triangles,  etc.  The  importance  of  the  circular 
functions  lies  in  the  power  they  possess  of  expressing  natural 
laws  of  a  periodic  character. 

70.  Rotation  of  Any  Locus.  In  §36  we  have  shown  that  any 
locus  y  =  f{x)  is  translated  a  distance  a  in  the  x  direction  by 
substituting  (x  —  a)  for  x  in  the  equation  of  the  locus.  Likewise 
the  substitution  of  (y  —  b)  for  y  was  found  to  translate  the  locus 
the  distance  b  in  the  y  direction.  A  discussion  of  the  rotation  of 
a  locus  was  not  considered  at  that  place,  because  a  displacement 
of  this  type  is  best  brought  about  when  the  equations  are  ex- 
pressed in  polar  coordinates. 

If  a  table  of  values  be  prepared  for  each  of  the  loci 

p  =  cos  8  (1) 

P  =  cos  (9i  -  30°)  (2) 

as  follows: 


134        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§70 


Equation  1 

Equation  2 

e 

p 

9i 

P 

-30° 

0.866 

0° 

0.866 

-20° 

0.940 

10° 

0.940 

-10° 

0.985 

20° 

0.985 

0° 

1.000 

30° 

1.000 

10° 

0.985 

40° 

0.985 

20° 

0.940 

50° 

0.940 

30° 

0.866 

60° 

0.866 

40° 

0.766 

70° 

0.766 

60° 

0.643 

80° 

0.643 

60° 

0.500 

90° 

0.500 

and  then  if  the  graph  of  each  be  drawn,  Fig.  70,  it  will  be  seen  that 
the  curves  differ  only  in  location  and  not  at  all  in  shape  or  size. 

If  a  value  be  given  to  61 
in  the  second  equation  which 
is  30°  greater  than  a  value 
given  to  d  in  the  first  equa- 
tion, the  two  values  of  p 
from  equations  (1)  and  (2) 
are  equal.  Thus,  if  AOP  is 
the  value  given  to  S  in  equa- 
tion (1)  and  if  AOP'  = 
AOP  +  30°  is  the  value 
given  to  Oi'va.  equation  (2), 
then  OP  will  equal  OP'. 
Thus  the  point  P'  may  be 

_,_„„.  ,     ,  looked  upon  as  having  been 

J)iG.  70. — Kotation    of    the    circle   „i  .   ;     j  f    „    .i  ■  i  o 

OAP    [p  =a  cos  8]   to   the  position  obtamed  from  the  pomt  P 

OA'P'  \p  =  a  cos  (e  —  30°)].  by  a  positive  rotation  about 

0  of  30°.  Thus  the  graph 
for  p  =  cos  {d  -  30°)  may  be  obtained  from  the  graph  for  p  = 
cos  d  by  rotating  it  about  the  pole  0  through  an  angle  of  30°. 

The  same  reasoning  will  apply  if  {fi  -  a)  be  substituted  for  6; 
in  this  case  the  locus  of  the  first  curve  is  rotated  about  the  pole 
through  an  angle  a,  in  the  positive  sense  if  a  be  positive,  in  a 
negative  sense  if  a  be  negative. 


§71]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   135 

By  the  same  reasoning  as  used  above,  we  see  that  if  in  the  polar 
equation  of  any  curve,  d  is  replaced  by  (6  —  a),  the  graph  of  the 
new  equation  is  the  graph  of  the  original  equation  rotated  about 
the  pole  through  an  angle  a,  but  is  otherwise  unchanged. 


Thboeems  on  Loci 

XIV.  If  {d  —  a)  be  substituted  for  6  throughout  the  polar  equa- 
tion of  any  locus,  the  curve  is  rotated  through  the  angle  a. 

Note  that  the  rotation  is  positive  when  a  is  positive  and  nega- 
tive when  a.  is  negative. 

Exercises 


paper 


draw:    a  =  cos  i 


1.  Upon   a  sheet  of   polar   coordinate 
p  =  cos  (9  -  60°);  p  =  cos  (9  +  60°). 

2.  Upon    a    sheet   of   polar   coordinate   paper   draw:   p  =  sin  B; 
p   =  sin  (.9  -  30°j;  p  =  sin  [6  '+  30°). 

3.  Upon    a    sheet    of   polar    coordinate   paper   draw:  p  =  cos  9; 

P  =  cos  (fl  —  I) ;  p  =  cos  (9  +  |j ;  p  =  cos  (9  —  t). 

71.  Polar  Equation  of  the  Straight  Line.    In  Fig.  71  let  MN  be 
any    straight    line  in  the 
plane  and  OT  be  the  per-  »^ 

pendicular  dropped  upon 
MN  from  the  pole  0.  Let 
the  length  of  OT  be  a  and 
let  the  direction  angle  of 
OT  be  a,  where,  for  a 
given  straight  line,  a  and 
a  are  constants.  Let  p  be 
the  radius  vector  of  any 
point  P  on  the  line  MN 
and  let  its  direction  angle 
be  d.     Then,  by  definition, 

-  =  cos  (6  —  or).  Fig.  71. — ^Equation  of  MN  is  a  =  p  cos 

P  (e-a). 

Therefore  the  equation  of  the  straight  line  MN  is 

a  =  p  cos  {d  —  a),  (1) 


136        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§72 

for  it  is  the  equation  satisfied  by  the  (p,  6)  of  every  point  of  the 
line.  This  is  the  equation  of  any  straight  line,  for  its  location  is 
perfectly  general.  The  constants  defining  the  line  are  the  per- 
pendicular distance  a  upon  the  given  line  from  0,  and  the  direction 
angle  a  of  this  perpendicular.  The  perpendicular  OT,  or  a,  is 
called  the  normal  to  the  line  MN,  and  the  equation  (1)  is  called 
the  normal  equation  of  the  straight  line. 
The  equation  of  the  circle  shown  in  the  figure  is 

Pi  =  o  cos  {6  —  a),  (2) 

in  which  pi  represents  the  radius  vector  of  a  point  Pi  on  the  circle. 
The  relation  pi  p  =  a^,  which  can  be  deduced  from  (1)  and  (2), 
is  interesting.    Because  of  it,  the  circle  is  often  called  the  inverse 
of  the  line  MN  with  respect  to  the  point  0. 

Exercises 

1.  Write  the  polar  equation  of  the  line  tangent  tp  the  circle  p  = 
5  cos  (9  —  30°)  at  the  end  of  the  diameter  passing  through  the  pole. 

2.  A  line  is  3  units  distant  from  the  pole  and  makes  an  angle  of 
45°  with  the  polar  axis.     Write  its  polar  equation. 

3.  Describe  the  curves  p  =  10  cos  I  *  —  4)  and  10  =  p  cos  ( ^  ~  i)  • 
Draw  the  following  circles : 


4. 

p   =  3  COE 

1  (e  -  30°). 

7.  P  = 

2  sin  (6  +  135°). 

6. 

p  =  3  cos 

{e  +  120°). 

8.  p  = 

^cos{e  +  l)- 

6. 

p  =  2  sin 

(9  -  45°). 

9.  p  = 

:  5  sin  (1  -  e)  • 

10. 

Show  that  p 

=  a  sin  9  is  the  locus  p 

=  0  cos  9  rotated  90°  counter 

clockwise. 
72.  Relation   between   Rectangular   and   Polar  Coordinates. 

Think  of  the  point  P,  Fig.  72,  whose  rectangular  coordinates  are 
(x,  y).  If  the  radius  vector  OP  be  called  p  and  its  direction  angle 
br  I'alhd  0,  then  the  polar  coordinates  of  P  are  (p,  6).  Then  x  and 
y  fo:  any  position  of  P  are  the  projections  of  p  through  the 
angle  6,  and  the  angle  (90°  —  d),  respectively,  or 

X  =  p  cos  d,  (1) 

y  =  p  sin  8.  (2) 


§72]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   137 


These  are  the  equations  of  transformation  that  enable  us  to  write 
the  equation  of  a  curve  in  polar  coordinates  when  its  equation 
in  rectangular  coordinates  is  known,  or  vice  versa.  Thus  the 
straight  line  x  =  3  has  the  equation 

p  cos  6  =  3 
in  polar  coordinates.    The  line  x  +  y  =  3  has  the  polar  equation 

p  cos  d  +  p  sin  6  =  3. 
The  circle  x^  +  y^  =  a^  has  the  equation 
p^cos"  9  +  p2  sin^  e  =  o^ 


or 
or 


To  solve  equations  (1)  and  (2) 
for  8,  we  write 

6  =  the  angle  whose  cosine  is  -> 

P 

6  =  the  angle  whoser  sine  is  -• 

P 

The    verbal    expressions     "the 

„„„!„  „,!,„„„  „„„; ;„  "   „i„     Fig.  72. — Rectangular  and  polar 

angle  whose  cosine  is,     etc.,  are         coordinates  of  a  point  P. 
abbreviated  in  mathematics  by 

the  notations  "arc  cos,"  read  "arc-cosine,"  and  "arc  sin,"  read 
"arc-sine,"  as  follows: 

6  =  aic  cos  (x/p)  (3) 

9  =  arc  sin  (y/p)  (4) 

Dividing  the  members  of  (2)  by  the  members  of  (1)  we  obtain 

y 

tan  0  =  -J  which,  solved  for  6,  we  write 


0  =  the  angle  whose  tangent  is 


y 


which  may  be  abbreviated 

d  =  arc  tan  (y/x) 

and  read  "8  =  the  arc-tangent  of  y/x." 
The  value  of  p  in  terms  of  x  and  y  is  readily  written 

P  =  VxM^- 


(5) 


(6) 


138        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§73 

Exercises 

1.  Write  the  polar  equation  of  x'  +  y'  +  8x  =  0. 
The  result  is  p*  +  8p  cos  fl  =  0,  or  p  =  —  8  cos  e. 

2.  Write  the  polar  equations  of  (a)  x'  +  y^  —  4j/  =0;  (6)  a;'  +  y' 
-  6x  -  iy  =  Q;  (c)  x^  +y'-  Qy  =  4. 

3.  Write  the  polar  equations  of  {a)  x  +  y  =  1;  (6)  x  +  2y  =  1; 
(c)  X  +  Vly  =  2. 

4.  Write  the  rectangular  equations  of  (a)  p  cos  0  +  p  sin  9  =  4; 
(6)  p  cos  e  —  3p  sin  9  =  6. 

5.  Write  the  polar  equation  of  x*  +  2y'  —  4x  =  0. 

6.  Write  the  rectangular  equation  of  p  =  2  cos  9  +  3  sin  9. 
Hint:  Multiply  both  members  of  the  equation  by  p,  replace  p'  by 

(x'  +  j/2),  p  cos  9  by  X,  and  p  sin  9  by  y. 

7.  Write  the  rectangular  equation  of  p  =  3  cos  9  —  2  sin  9. 

8.  Write  the  rectangular  equation  of  p  =  5  sin  9  —  3  cos  9. 

73.  Identities  and  Conditional  Equations.    It  is  useful  to  make 
a  distinction  between  equalities  like 

(a  -  x){a  +  x)  =  a'  -  x\  (1) 

which  are  true  for  all  values  of  the  variable  x;  and  equalities  like 

x^  -2x  =  3,  (2) 

which  are  true  only  for  certain  particular  values  of  the  unknown 
number.  When  two  expressions  are  equal  for  all  values  of  the 
variable  for  which  the  expressions  are  defined,  the  equality  is 
known  as  an  identity.  When  two  expressions  are  equal  only  for 
certain  particular  values  of  the  unknown  number,  the  equality  is 
spoken  of  as  a  conditional  equation.     The  fundamental  formula 

sin^  (j)  +  cos''  0  =  1 
is  an  identity. 

2  sin  A  +  3  cos  ^  =  3.55 

is  a  conditional  equation.  The  symbol  =  is  sometimes  used  to 
distinguish  an  identity;  thus 

a'  —  a;'  =  (a  —  x){a'^  -\-  ax  +  x''). 

The  following  illustrations  and  exercises  contain  problems  both 
in  the  establishment  of  trigonometric  identities  and  in  the  finding  of 
the  values  of  the  unknown  number  from  trigonometric  conditional 
equations. 


§73]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   130 

The  truth  of  a  trigonometric  identity  may  be  established  by 
reducing  each  side  to  the  same  expression.  In  this  work,  however,  the 
student  will  be  required  to  transform  the  left-hand  side  by  means  of  the 
fundamental  relations  (2)  to  (6),  §64,  until  it  is  identically  equal  to  the 
right-hand  side. 

Facility  in  the  establishment  of  trigonometric  identities  is  largely  a 
matter  of  skill  in  recognizing  the  fundamental  forms  and  of  ingenuity 
in  performing  transformations.  All  solutions  of  conditional  equations 
should  he  checked.  The  following  worked  exercises  will  illustrate  the 
method. 

Illustration  1:  Show  that  (1  —  sin  u  cos  u)  (sin  u  +  cos  u)  ^ 
sin'  u  +  cos'  u.     Taking  the  left-hand  member 

(1  —  sin  u  cos  u)  (sin  u  +  cos  u) 

=  sin  u  +  cos  w  —  sin^  u  cos  u  —  sin  u  cos^  u 
=  cos  u  {1  —  sin^  m)  +  sin  u  (1  —  cos'  u) 
=  cos  u  cos'  u  +  sin  u  Bin'  u 
=  cos'  u  +  sin'  u. 

This  last  expression  is  the  right-hand  member  of  the  given  identity. 
Thus  the  identity  is  verified. 
Illustration  2:  Show   that  sec' a;  —  1   =  sec'  x  sin'  x. 

sec' a;  —  1  =  see's  (1 -; — )  =  sec'x  (1  —  cos'  x)  =  see's  sin's. 

\       sec'  x/ 

Illustration  3:  Solve  for  all  values  of  x  less  than  360°    ' 

2  sin  X  +  cos  s  =  2. 

Transposing  and  squaring  we  get 

cos'  X  =4  —  8sina;-|-4  sin'  x. 

Since  sin'  x  +  cos'  s  =  1, 

1  —  sin'  s  =  4  —  8  sin  x  +  4  sin'  x, 
5  sin'  a;-8sins  +  3=0, 
sin  s  =  1  or  0.6 
X  =  90°,  and  37°  or  143°  approximately. 
Check:  2  sin  90°  -t-  cos  90°  =  2  -|-  0  =  2 

Check:  2  sin  37°  -f-  cos  37°  =  1.2  +  0.8  =  2 

Does  2  sin  143°  +  cos  143°  =  1.2  -  0.8  =  0.4  =  2? 

The  last  value  does  not  check.  The  reasons  for  this  will  be  dis- 
cussed later  in  §98.  Therefore  the  correct  solutions  are  90°  and  37° 
approximately. 


140        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§74 


Exercises 

1.  Solve  6  cos"  e  +  5  sin  9  =  7  for  all  values  of  ff  <  90°. 
Suggestion:  Write  6(1   —  sin*  9)  +  5  sin  9  =  7  and  solve  the 

quadratic  in  sin  9. 

6  sin"  9-5  sin  9  +  1  =0, 
or 

(3  sin  9  -  1)(2  sin  9  -  1)  =  0. 

sin  9  =  i  or  J 

0  =  19.6°  approximately  and  30°. 

The  results  should  be  checked. 

2.  Prove  that  for  all  values  of  9  (except  ir/2  and  3ir/2,  for  which 
the  expressions  are  not  defined) 

sec*  9  —  tan'  9  =  tan*  9  +  sec*  9. 

3.  Show  that  ' 

sec'  u  —  sin*  u  =  tan*  u  +  cos*  u, 

for  all  values  of  the  variable  u  except  90°  and  270°,  for  which  the 
expressions  are  not  defined. 

4.  Find  u,  if  tan  u  +  cot  u  =  2. 

6.  Find  sec  9,  if  2  cos  9  +  sin  9  =  2. 

6.  Show  that 

sec  a  +1  tana 

tan  a     ~  sec  o  —  1 

Hint:  Multiply  both  numerator  and  denominator  of  the  left-hand 
member  by  (sec  a  —  1). 

7.  Show  that  sec  a  +  tan  a  = x 

sec  a  —  tan  a 

8.  Show  that  sin*  a  +  sin*  a  tan*  a  =  tan*  a. 

9.  Show  that  (esc*  a  —  1)  sin*  a  =  cos*  a. 

10.  Show  that 

sin  A      _  1  +  cos  A 
1  —  cos  A  ~     sin  A 

11.  Show  that  2  cos*  m  —  1  =  cos*  u  —  sin'  u. 

12.  Show  that  cos'  a  —  sin'  a  +  1  =  2  cos*  a. 

13.  Show  that  sec*  u  +  esc*  u  =  esc*  u  sec*  u. 

14.  Show  that  (tan  a  +  cot  o)*  =  sec*  a  esc*  a. 

16.  Solve  sec  x  —  tan  s  +  1  =  0  for  all  values  of  x  less  than  360°. 
74.  The  Graph  of  p  =  a  cos  0  +  b  sin  0.    Before  reading  this 
section  the  student  should  review  exercises  6  and  7,  §72.    Let  us 


§74]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   141 


find  the  Cartesian  equation  for  the  curve  whose  polar  equation  is 

p  =  a  cos  0  +  6  sin  6,  (1) 

where  a  and  b  are  any  constants,  positive  or  negative.    First 
multiply  each  member  of  (1)  by  p. 


ap  cos  B  +  bp  sin  6 


(2) 


Since  p^  =  x^  +  y^,  p  cos  B  =  x,  and  p  sin  0  =  y,  equation  (2) 
beaomes 

x^  +  y^  =  ax  +  by.  (3) 

Transposing  and  completing  squares 


[■-ir-b-i]' 


+  b' 


(4) 


Fio.  73. — The   circles   p  =  a  cos  e,   p  =  b  sin  $,   and   p  =  a  cos  9  + 
6  sin  e,  or  the  circles  OA,  OB,  and  OC  respectively. 

This  is  the  Cartesian  equation  of  a  circle  with  center  at  the  point 
(io,  ib)  and  of  radius  iy/a'  +  6^.  The  circle  passes  through  the 
pole  or  origin  since  the  coordinates  (0,  0)  satisfy  the  equation 
(3),  and  also  passes  through  the  point  (a,  6),  since  these  coordiT 
nates  satisfy  (3).  Thus  if  upon  the  diameters  of  the  circles  p  = 
o  cos  6  and|P  =  6  sin  B,  we  construct  a  rectangle,  the  circle  having 
a  diagonal  of  this  rectangle  as  a  diameter,  is  the  locus  of  p  = 
aoosd  +  b  sin  B.    See  Fig.  73. 


142        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§75 

Exercises 

Draw  the  graphs  for  the  following : 

1.  p  =  2  cos  9  +  2  sin  9.  2.  p  =  3  cos  9  +  2  sin  9. 

3.  p  =  —  2  cos  9  +  2  sin  9.  4.  p  =  —  3  cos  9  —  2  sin  9. 

5.  In  Fig.  74  let  a  =2  and  a  =  30°.     Find  the  equation  of  each  of 
the  four  circles  in  the  form  p  =  a  cos  9  +  6  sin  9. 


p=ac°^^ 
Fig.  74. — Diagram  for  Exercise  5. 

75.  Additive  Properties.  The  shearing  of  a  curve  in  a  straight 
line,  considered  in  §38,  may  be  thought  of  as  the  addition  of  the 
ordinates  of  the  curve  and  of  the  straight  line,  corresponding  to  a 
given  value  of  the  abscissa.  This  sum  is  the  corresponding 
ordinate  of  the  new  curve.  In  the  more  general  case  the  curve 
y  =  fW)  +  P'(x)  may  be  constructed  from  the  curves  y  =  f(x) 
and  y  =  F{x)    by  adding  their  ordinates.     Thus  the  curve  for 

y  =  x^-\ — )  Fig.  75,  was  constructed  by  adding  the  ordinates  of 

the  curves  y  =  x^  and  j/  =    • 

In  the  same  way  the  curve  for  p  =  f(d)  +  F{6)  may  be  con- 
structed from  the  curves  p  =  f{d),  and  p  =  F(,d)  by  adding  the 
radius  vectors  corresponding  to  the  same  value  of  the.  vectorial 
angle.  Thus  points  on  the  circle  p  =  2  cos  6  +  3  sin  6,  Fig.  73, 
may  be  located  by  adding  (using  the  bow  dividers)  the  radius 


§76]     THE  CIECLE  AND  THE  CIRCULAR  FUNCTIONS   143 

vectors  of  p  =  2  cos  9  and  p  =  3  sin  6.    That  is,  OP  =  OPi  +  OPi 
for  all  positions  of  OP. 

I  Exercises 

1.  Plot  on  polar  coordinate  paper  the  curve  for  p  =3  cos  9  +  2  sin  9 
making  use  of  the  circles  p  =  3  cos  9  and  p  =  2  sin  9. 

2.  Plot  on  polar  coordinate  paper  the  curve  for  p  =  cos  9+1 
making  use  of  the  circles  p  =  cos  9,  and  p  =  1.     Note  that  when 


90°  <  9  <  270°,  the  p  for  p  =  cos  9  is  negative,  and  that  the  addition 
referred  to  above  is  algebraic  addition. 

3.  Plot  upon  polar  coordinate  paper  the  curve  for  p  =  1  +  sin  9, 
making  use  of  the  circles  p  =  1,  and  p  =  sin  9. 

4.  Plot  upon  polar  coordinate  paper  the  curve  for  p  =  2  cos  9  —  1. 
,6.  Plot  upon  polar  coordinate  paper  the  curve  for  p  =  cos  9  +  2. 

76.  Graph  of  y  =  tan  x.    If  this  graph  is  to  be  constructed  on 
a  sheet  of  ordinary  letter  paper,  8^  inches  X  U  inches,  it  is  desirable 


144        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§76 


to  proceed  as  follows:*  Draw  at  the  left  of  the  sheet  of  paper  a  semi- 
circle of  radius  1.15  .  .  inches  (that  is,  of  radius  =  18/5ir),  so 
that  the  length  of  the  arc  of  an  angle  of  10°,  or  ir/18,  radians  will  be 
i  of  an  inch.  Take  for  the  X-axis  a  radius  COX  prolonged,  and 
take  for  the  F-axis  the  tangent  OY  drawn  through  0,  as  in  Fig. 


r                  MA                                              M' a' 

/T\J lL_______ 

/A  W                 \/ 

Q 

rUj/T        I                 _          _   _      E        

k//^___l^\ /_\ 

T^ZA      Z        5                      _-_   ^Z        ^^ 

/^^5Z           ^                        z           S^   ^ 

l^^—/                     \                           /                     K    ^ 

fe:cr— -P                  -^^•-                   ^'^                -il^                   ^ 

^^^^  '                  "    "^             Z                    3^             Z 

wxSc                    ~^,       Z                         \  ^  I 

VV>^                    _S^l "_Z___ 

^\ t :_j____ 

\\                /  '\                    /  \ 

\     /  '^      w 

2T-" 


Y'  B     N'  B'  N" 

Fig.  76. — Graphical  construction  of  the  curve  of  tangents  y  = 
tan  X.  For  lack  of  room  only  a  few  of  the  points  Si,  St,  ...  Ti,  Ti, 
. .  .are  lettered  in  the  diagram.    The  dotted  curve  is  s/  =  cot  x. 

76.  Divide  the  semicircle  into  eighteen  equal  parts  and  draw  radii 
through  the  points  of  division  and  prolong  them  to  meet  OF  in 
points  Ti,  Ti,  Ti,  Tt,  .  .  .  Then  on  the  F-axis  there  is  laid  off 
a  scale  YY'  in  which  the  distances  OTi,  OT2,  .  .  are  propor- 
'  tional  to  the  tangents  of  the  angles  OCSi,  OCS2,  .  .  . ;  for  the 
tangents  of  these  angles  are  OTi/CO,  Orj/CO,  .  .  .  and  CO  is 
the  unit  of  measure  made  use  of  throughout  this  diagram.  Draw 
horizontal  lines  through  the  points  of  division  on  OF  and  vertical 

>  The  student  should  understand  the  construction  of  Figs.  76  and  78,  but  it  is 
opt  necessary  that  be  actually  draw  them. 


[§77]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   145 

lines  through  J  inch  intervals  on  OX,  thus  dividing  the  plane 
into  a  large  number  of  small  rectangles.  Starting  at  0,  t,  2fir, 
...    —  IT,   —  2ir,  .  and  sketching  the  diagonals  of  con- 

secutive cornering  rectangles,  the  curve  oi  y  —  tan  x  is  approxi- 
mated. Greater  precision  may  be  obtained  by  increasing  as 
desired  the  number  of  divisions  of  the'  circle  and  the  number  of 
corresponding  vertical  and  hprizontal  lines. 

It  is  observed  that  the  graph  of  the  tangent  is  a  series  of  similar 
branches,  which  are  discontinuous  for  x  =  ir/2,  —  ir/2,  (3/2)ir, 
—  (3/2)ir,  ...  At  these  values  of  x  the  curve  has  vertical 
asymptotes,  as  shown  at  AB,  A'B',  in  Fig.  76. 

If  the  number  of  corresponding  vertical  and  horizontal  lines 
be  increased  sufficiently,  the  slope  of  the  diagonal  of  any  rectangle 
gives  a  close  approximation  to  the  true  slope  of  the  curve  at  that 
point. 

It  has  already  been  noted  that  all  of  the  trigonometric  functions 
are  periodic  functions  of  period  2v.  It  is  seen  in  this  case,  how- 
ever, that  tan  x  has  also  the  shorter  period  x;  for  the  pattern 
M'N'  of  Fig.  76  is  repeated  for^each  interval  ir  of  the  variable  x. 

77.  Graph  of  cot  x.  In  order  to  lay  off  a  sequence  of  values  of 
cot  S  on  a  scale,  it  is  convenient  to  keep  the  denominator  con- 


P»   Ps  Pi 


Pr,  P.    P^ 


Pi 


^ 

^ 

7 

m 

\M 

j£ 

4 

^ 

^ 

Pii  Z>io     DsDtD,         O         DiDi    Dz        Di  Di 

Fig.  77. — Construction  of  a  scale  of  cotangents. 


stant  in  the  ratio  (abscissa) /(ordinate)  which  defines  the  cotangent. 
The  denominator  may  also,  for  convenience,  be  taken  equal  to 
unity.    Thus,  in  Fig.  77,  the  triangles  of  reference  DiOPi,  D2OP2, 

for  the  various  values  of  6  shown,  have  been  drawn  so  that 
the  ordinates  P\Di,  PzDi,  .  are  equal.    If  the  constant  ordi- 

nate be  also  the  unit  of  measure,  then  the  sequence    ODi,  OD2,  OD3 , 

ODt,  ODi,  represents, '  in  magnitude  and  sign,  the  cotan- 
gents of  the  various  values  of  the  argument  d.  Using  ODi,  OD2, 
...  as  the  successive  ordinates  and  the  circular  measure  of  Q 

10 


146        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§78 

as  the  successive  abscissas,  the  graph  oi  y  —  cot  x  is  drawn,  as 
shown  by  the  dotted  curve  in  Fig.  76. 

The  sequence  ODi,  OD2,  Fig.  77  is  exactly  the  same  as 

the  sequence   OTi,   OT2,    .  Fig.   76,   but   arranged  in  the 

reverse  order.  Hence,  the  graph  of  the  cotangent  and  of  the 
tangent  are  alike  in  general  form,  but  one  curve  descends  as  the 
other  ascends,  so  that  the  position,  in  the  plane  ZF,  of  the  branches 
of  the  curve  are  quite  different.  In  fact,  if  the  curve  of  the 
tangents  be  rotated  about  07  as  axis  and  then  translated  to  the 
right  the  distance  ir/2,  the  curves  would  become  identical. 
Therefore,  for  all  values  of  x, 

tan  (7r/2  —  x)  =  cot  x.  (1) 

This  is  a  result  previously  known. 

78.  Graph  of  y  =  sec  x.  Since  sec  6  is  the  ratio  of  the  radius 
divided  by  the  abscissa  of  any  point  on  the  terminal  side  of  the 
angle  d,  it  is  desirable,  in  laying  off  a  scale  of  a  sequence  of  values 
of  sec  d,  to  draw  a  series  of  triangles  of  reference  with  the  abscissas 
in  all  cases  the  same,  as  shown  in  Fig.  78.  In  this  figure  the  angles 
were  laid  off  from  CQ  as  initial  line.    Thus 

CTs/CSi  =  sec  QCSi, 

or,  if  CSi  be  unity,  the  distances  like  CTt,  laid  off  on  CQ,  are  the 
secants  of  the  angles  laid  off  on  the  arc  QSi,0  or  laid  off  on  the  axis 
OX. 

The  student  may  describe  the  manner  in  which  the  rectangles 
made  by  drawing  horizontal  lines  through  the  points  of  division  on 
CQ  and  the  vertical  Unes  drawn  at  equal  intervals  aloiig  OX,  may 
be  used  to  construct  the  curve.  If  the  radius  of  the  circle  be  1.15 
inches,  what  should  be  the  length  of  Oir  in  inches? 

The  student  may  sketch  the  locus  oi  y  =  esc  x,  and  compare 
with  the  locus  y  —  sec  x. 

Exercises 

1.  Discuss  from  the  diagrams,  59,  76,  78,  the  following  statements: 
Any  number,  however  large  or  small,  is  the  tangent  of  some  angle. 
The  sine  or  cosine  of  any  angle  cannot  exceed  1  in  numerical  value. 
The  secant  or  cosecant  of  any  angle  is  always  numerically  greater^ 
than  I  (,or  at  least  equal  to  1), 


§79]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   147 

2.  Show  that  sec  (o  ~  ^)  ~  <'*°  ^  ^°''  ^^  values  of  x. 

3.  If  tan  9  sec  9  =  1,  show  that  sin  $  =  KVs  —  1)  and  find  9 
by  use  of  polar  coordinate  paper,  Form  M3. 

4.  Describe  fully  the  following,  locating  nodes,  troughs,  crests,  etc. : 

(a)  y  =  sin  [x  -"^y  (c)  y  =  tan  y>  +lj' 

(b)  y  =  cos  (^J  + 1)  '  (d)  y  =  tan  (x  +  1). 


r 

N 

A 

N' 

\\ 

/ 

1 

\ 

T 

\\\ 

/ 

^ 

\  \  \  \ 

/p 

S         '~ 

WW  \ 

/ 

_ 

._ 

^ 

« 

®„ 

m 

7 

r 

M 

2' 

'///I 



7 

< 



" 

/// 

/ 

\ 

/■/  / 

/ 

\ 

/  / 

/ 

\ 

I"  B    N'  ,  N' 

Fig.  78. — Graphical  construction  of  y'  =  sec  x. 


79.  Increasing  and  Decreasing  Functions.  The  meanings  of 
these  terms'  have  been  explained  in  §27.  Applying  these  terms  to 
the  circular  functions,  we  may  say  that  y  =  sin  x,  y  =  tan  x, 
y  =  sec  X  are  increasing  functions  for  0  <  a:  <  ■k/2.  The  co- 
functions,  y  =  cos  z,  y  —  cot  x,  y  =  esc  x,  are  decreasing  func- 
tions within  the  same  interval. 


148       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§79 

Exercises 

Discuss  tlie  following  topics  from  a  consideration  of  the  graphs  of 
the  functions: 

1.  In  which  quadrants  is  the  sine  an  increasing  function  of  the 
angle?     In  which  a  decreasing  function? 

2.  In  which  quadrants  is  the  tangent  an  increasing,  and  in  which  a 
decreasing,  function  of  its  variable? 

3.  In  which  quadrants  are  the  cos  0,  cot  6,  sec  9,  esc  6,  increasing 
and  in  which  are  they  decreasing  functions  of  9? 

4.  Show  that  all  the  co-functions  of  angles  of  the  first  quadrant  are 
decreasing  functions. 


1.  Show  that 

2.  Show  that 


Miscellaneous  Exercises 

tan' a  .  , 

T — r^ — r-  —  sin'  a 
1  +  tan' a 

\/l  —  sin'a  COS  a 


y/\  —  COS 


3.  Show  that  cot'  a  —  cos'  a  =  cot'  a  cos*  o. 

4.  Show  that 

s  Vcsc'  -  1* 


6.  Show  that 

6.  Show  that 

7.  Show  that 


Vsec'a  —  1 

1  +  tan'  a  __  sin'  a 
1  +  cot'  a.  "  cos'  a 


1  +  cos  a 
8.  Show  that 

CSC  a 


\\  —  sin  a  , 

\'- — - — , =  sec  a  —  tan  a. 
I  +  sin  a 

sin  a        ,   1  +  cos  a 

-\ ; =  2  CSC  a 


COS  a. 


cot  a  +  tan  a 

9.  Show  that 

1 

— T r-L =  sin  u  cos  M. 

cot  u  +  tan  u 

10.  Show  that 

CBO*  M  (1  -r  cos*  m)  —  2  cot'  Mai. 


§79]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    149 

11.  Find  the  distance  of  the  end  of  the  diameter  of  , 

p  =  8  cos  (9  -  60°) 
from  the  line  OX. 

12.  If  PI  =  a  cos  9,  and  P2  =  a  sin  6,  find  pi  —  pi  when  0  =  60° 
and  0=5. 

13.  Find  the  polar  equation  of  the  circle  x'  +  y'  +  Qx  =  0: 

14.  For  what  value  of  9  does  p  =  3.55,  if  p  =  2  sin  9  +  3  cos  9? 
Result:  9  =  23°  30'  and  43°  30'.    Hint:  Draw  the  circles  p  =  3.55 

and  p  =  2  sin  9  +  3  cos  9  on  polar  coordinate  paper  and  find  the 
vectorial  angles  for  the  two  points  of  intersection.  This  problem  is 
the  same  as:  "Solve  the  equation  2  sin  9  +  3  cos  9  =  3.55  for  9." 

16.  Solve  graphically  the  equation  2  sin  9  +  3  cos  9  =  2. 

Hint:  Draw  on  polar  coordinate  paper  the  curves  p  =  2  and 
p  =  2  sin  9  +  3  cos  9. 

16.  Solve  graphically  the  equation  4  cos  9  —  3  sin  9=  3.5. 

17.  Find  sin  9  if  esc  9=  vV_±A'. 

a 

18.  A  circular  arc  is  4,81  inches  long.  The  radius  is  12  inches. 
What  angle  is  subtended  by  the  arc  at  the  center?  Give  result  in 
radians  and  in  degrees. 

19.  Certain  lake  shore  lots  are  bounded  by  north  and  south  lines 
66  feet  apart.  How  many  feet  of  lake  shore  to  each  lot  if  the  shore- 
line is  straight  and  runs  77°  30'  E.  of  N.7 

20.  If  2/  =  2  sin  A  +  3  cos  A  -  3.55,  take  A  as  20°;  as  23°;  as  26° 
and  find  in  each  case  the  value  of  y.  From  the  values  of  y  just 
found  find  a  value  of  A  for  which  y  is  approximately  zero.  This 
process  is  known  as  "cut  and  try." 

21.  The  line  y  =  ^x  ia  to  coincide  with  the  diameter  of  the  circle 
p  =  10  cos  (9  —  a).    Find  a. 

22.  The  line  y  =  2x  is  to  coincide  with  the  diameter  of  the  circle 
p  =  10  sin  (9  +  a).     Find  a. 

23.  To  measure  the  width  of  the  slide  dovetail  shown  in  Fig.  79, 
two  carefully  ground  cylindrical  gauges  of  standard  dimensions  are 
placed  in  the  V'a  at  A  and  B,  as  shown,  and  the  distance  X  carefully 
taken  with  a  micrometer.  The  angle  of  the  dovetail  is  60°.  Find 
the  reading  of  the  micrometer  when  the  piece  is  planed  to  the  required 
dimension  MN  =  4  inches.  Also  find  the  distance  Y.  (Adapted 
from  "Machinery,"  N.  Y.) 

24.  Sketch  y  =  ix and  y  =  sinx  and  then  y  =  ix  —  sinx. 

26.  Sketch  the  curve  y  =  cos  .-c  +  2  sin  x,  making  use  of  the  curves 
y  =  cos  X  and  ^  =  2  sin  x. 


150        ELEMENTARY  MATHEMATICAL  ANALYSIS        [|79 

26.  Find  the  maximum  value  of  the  function  given  in  exercise  25. 
Hint:  Find  the  maximum  value  of  p  in  the  graph  of  p  =  cos  9 

+  2  sin  e. 

27.  Find  the  maximum  value  of  2  cos  x  —  3  sin  x. 

28.  Since  p  =  cos  9  +  2  sin  fl  is  a  circle  passing  through  the  pole, 
the  equation  may  be  put  in  the  form  p  ='  a  cos  (9  —  a).  Find  a 
and  a. 

Result:  o  =  VS  and  a  =  63°  20'  approximately. 

29.  A  circle  is  inscribed  in  a  30°,  60°  right  triangle.  Find  the  diame- 
ter of  the  circle  (a)  if  the  shorter  leg  of  the  triangle  is  2  inches;  (6) 
if  the  longer  leg  is  2  inches;  (c)  if  the  hypotenuse  is  4  inches,  (d)  Find 
the  length  of  the  sides  of  the  triangle  if  the  diameter  of  the  inscribed 
circle  is  2  inches. 


Fig.  79. — Diagram  to  Exercise  23. 

30.  A  circle  is  inscribed  in  a  45°  right  triangle.     Find  the  diameter 
of  the  circle  if  the  legs  of  the  triangle  are  4  inches. 

31.  The  center  of  the  circle  p'=  10  cos  (9  —  a)  Ues  on  the  Ijne 
Zx  —  2y  =  \.     Find  two  possible  values  for  a. 

32.  The  center  of  the  circle  p  =  10  sin  (9  +  a)  lies  on  the  line 
X  —  22/  =  6.    JFind  two  possible  values  for  a. 

33.  Write  the  Cartesian  equations  for: 

(a)  p  =  2  cos  9  +  3  sin  9.  (b)  p  =  2  cos  9  —  5  sin  9. 

(c)  p  =  2  sin  9  —  5  cos  9. 

34.  Find  the  co6rdinates  of  the  center  and  the  radius  for: 

(o)  x»  +  2/2  -  2x  -  4i/  +  4  =  0     (d)  2x2  +  2y^  +  3x  +  by=  0 
(5)  x«  +  2/'  +  2x  +  42/  +  4  =  0     (e)   3x2  +  ^yi  _  gj.  -  ■y/2y  =  10 
(c)   x'  +  !/»  +  3x  -  42/  =  0  (/)   x2  +  ^2  +  7x-  \^Zy  =  25 


§79]     THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   151 

36.  Which  circles  of  exercise  34  pass  through  the  origin? 

36.  Write  the  equation  of  a  line  passing  through  the  origin  and  the 
center  of  the  circle  x'  +  y^  —  3x  —  5y  =  6. 

37.  Write  the  equation  of  a  Une  parallel  to3x  —2y  =  Q  and  passing 
through  the  center  of  x'  +  y''  —  Sx  —  2y  =  0. 


CHAPTER  V 


THE  ELLIPSE  AND  HYPERBOLA 

80.  The  Ellipse.  If  all  ordinates  of  a  circle  be  shortened  by 
the  same  fractional  amount  of  their  length,  the  resulting  curve 
is  called  an  ellipse.  For  example,  in  Fig.  80,  the  middle  points 
of  the  positive  and  negative  ordinates  of  the  large  circle  were 

marked    and    a    curve    drawn 
through  the  points  so  selected. 
The  result  is  the  ellipse 
ABA'B'A. 
If 

a;2  +  2/2  =  a*  (1) 

is  the  equation  of  a  circle,  then 

x^  +  {myY  =  o^         (2) 

in  which  m  is  any  constant  >  1, 
is  the  equation  of  an  ellipse; 
for  substituting  my  for  y  divides 
all  of  the  ordinates  by  m,  by 
Theorem  IX  on  Loci,  §28. 


Fig.  80.- 


-Construction  of  an 
ellipse. 


Dividing  both  members  of  (2)  by  a'  we  obtain. 


-i  +  '-iV 


=  1. 


(3) 


Let  6'  be  written  in  place  of  — ^-    Equation  (3)  becomes 


m'- 


V 
+  h  =  ^ 


(4) 


which  is  the  standard  form  of  the  equation  of  an  ellipse. 

81.  Orthographic  Projection  of  a   Circle.    The   ellipse  may 
also  be  looked  upon  as  the  orthographic  projection  of  the  circle. 

152 


§81] 


THE  ELLIPSE  AND  HYPERBOLA 


153 


Let  ABCD,  Fig.  81a,  be  a  circle  with  a  radius  o.  Let  AOC,  Fig. 
816,  be  an  end  elevation  of  the  same  circle.  Rotate  this  circle 
about  BOD  as  an  axis  through  an  angle,  /J,  to  the  position  A"OC". 
Project  the  rotated  circle  upon  its  original  plane,  into  the  curve 
A'BC'D.  We  shaU  show  that  A'BC'D  is  an  ellipse.  Take  any 
point  P  upon  the  original  circle.  It  rotates  into  the  point  P",  and 
P"  projects  into  P'.  The  equation  of  the  circle  is  x^  +  y^  =  o", 
where  y  =  MP.  To  get  the  equation  for  the  curve  A'BC'D 
replace  MP  by  its  equal  MP'/cos  /?.     (See  Fig.  81b.)    Whence, 


'  + 


(MP'y 

cos''/3 


Since  MP'  is  the  ^/-coordinate  of  P', 


x^  + 


y2 


cos'/3 


A 

P                 / 

] 

^ 

p 

P'            /^ 

~~?' 

^ 

\ 

n     ^\ 

o 

V 

a 

^\    VV 

/ 

/ 

y 

b    "  ~  a 

Fig.  81. — The  ellipse  considered  as  the  orthographic  projection  of 

a  circle. 


or 


V' 


o'      a^oos'jS 
Replacing  o  cos  jS  by  6(=  OC), 

the  equation  of  an  ellipse. 


1, 


As  a  consequence  of  the  above,  it  ia  seen  that  the  shadow  cast  on 


164        ELEMENTAKY  MATHEMATICAL  ANALYSIS        [§82 

the  floor  by  a  circular  hoop  held  at  any  angle  in  the  path  of  vertical 
rays  of  light  is  an  ellipse. 

If  the  abscissas  of  a  circle  be  lengthened  by  amounts  propor- 
tional to  their  lengths,  the  resulting  curve  is  an  ellipse.  Let 
x^  +  y^  —  V  be  the  equation  of  the  circle.    Then 

2 

+  2/2  =  b2 


is  the  equation  formed  by  lengthening  all  abscissas  in  the  raticf 
1 :  m,  TO  >  1.     Dividing  by  Ji^  and  replacing  m'-h'^  by  a},  we  obtain 

a^  "^  62       ^• 

Thus  the  ellipse  of  Fig.  80  could  have  been  formed  by  doubling  all 
of  the  abscissas  of  the  circle  BD'B'D.  Hence  we  see  that  if  all 
parallel  chords  of  a  circle  are  lengthened  or  shortened  by  an 
amount  proportional  to  their  length,  an  ellipse  is  formed.  If  the 
deformation  takes  place  in  chords  parallel  to  either  the  X-  or 
F-axis  the  equation  is  of  the  form  (4)  of  the  last  section,  called  the 
symmetric  equation  of  the  ellipse. 

The  diameter  of  the  circle  from  which  the  ellipse  may  be  formed 
by  shortening  parallel  chords  is  called  the  major  axis  of  the 
ellipse.  Thus  AA',  or  2a,  Fig.  80  is  the  major  axis  of  ABA'B'. 
The  diameter  of  the  circle  from  which  the  ellipse  could  have  been 
formed  by  lengthening  parallel  chords  is  called  the  minor  axis  of 
the  ellipse.  Thus  BB',  or  26,  Fig.  80  is  the  minor  axis  of  ABA'B'. 
The  point  of  intersection  of  the  axes  is  called  the  center  of 
the  ellipse.  One-half  of  the  major  and  minor  axes  are  called, 
respectively,  the  semi-major  and  semi-minor  axes  of  the  ellipse. 
The  points  A  and  A'  are  called  the  vertices  of  the  ellipse. 

82.  Ejcplicit  Form  of  Equation.    The  equation  of  the  ellipse 

o"  ^  62  ^'■' 

when  solved  for  y  may  be  put  in  the  important  form 

.y  =  +  -  Va^-x^  (2) 

The  equation  of  the  circle  x^  +  y^  =  a^  solved  for  y  is 

y  =  ±  Va^  -  x2-  (3) 


§83] 


THE  ELLIPSE  AND  HYPERBOLA 


155 


Equations  (2)  and  (3)  are  in  a  form  very  useful  for  many  purposes. 
It  is  easy  to  see  that  (2)  states  that  the  ordinates  of  the  ellipse  are 
the  fractional  amount  hfa  of  the  ordinates  of  the  circle -(3) . 

The  definition  of  the  term  function  permits  us  to  speak  of  j/  as  a 
function  of  x,  or  of  a;  as  a  function  of  y,  in  cases  like  equation  (1) 
above;  for  when  x  is  given,  y  is  determined.  To  distinguish  this 
from  the  case  in  which  the  equation  is  solved  for  y,  as  in  (2),  y,  in 
the  former  case,  is  said  to  be  an  implicit  function  of  x,  and  in  the 
latter  case,  y  is  said  to  be  an  explicit  function  of  x. 

83.  Section  of  a  Cylinder.  If  a  circular  cylinder  be  cut  by  a 
plane,  the  section  of  the  cylinder  is  an  ellipse.  For,  select  any 
diameter  of  a  circular  section  of 

the  cylinder  as  the  X-axis.  Let 
a  plane  be  passed  through  this 
diameter  making  an  angle  a 
with  the  circular  section.  Then 
if  ordinates  (or  chords  perpen- 
dicular to  the  common  X-axis) 
be  drawn  in  each  of  the  two 
planes,  all  ordinates  of  the  sec- 
tion made  by  the  cutting  plane 
can  be  made  from  the  ordinates 
of  the  circular  section  by  multi- 
plying them  by  sec  a.  Hence 
any  plane  section  of  a  cylinder 
is  an  ellipse. 

84.  Parametric  Equations  of  the  Ellipse.  Let  ABA'B'A,  Fig. 
82,  be  an  ellipse  whose  semi-major  axis  is  a  and  whose  semi-minor 
axis  is  6.  Upon  AA'  and  BB'  as  diameters  construct  circles. 
These  circles  are  called,  respectively,  the  major  and  minor  auxiliary 
circles.  From  the  origin,  draw  any  radius  vector,  as  QP2P1, 
making  an  angle  d  with  the  positive  direction  of  the  axis  of  x. 
Through  P2  and  Pi  draw  lines  parallel,  respectively,  to  the  X-  and 
F-axes,  and  let  P  be  their  point  of  intersection.  It  will  be  shown 
that  P  is  a  point  upon  the  ellipse. 

Let  the  coordinates  of  P  be  a;  and  y.    Then 


-A  construction  of  the 
ellipse. 


X  =  a  cos 


(1) 


156        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§84 
and 


or 


and 


Then 


y  =  b  sin  6,  (2) 


-  =  cos  8, 
a 


f  =sine. 

0 


^  +  f"  =  cos^  fl +sin2  e  =  1, 

which  shows  that  P  is  upon  the  ellipse. 

Equations  (1)  and  (2)  are  called  parametric  equations  of  the 
ellipse.    6  is  called  the  variable  parameter,  or  the  eccentric  angle. 

The  method  used  above  of  locating  points  upon  the  ellipse 
constitutes  one  of  the  best  practical  methods  of  constructing  an 
ellipse  when  its  axes  are  known.  For,  by  it  a  large  number  of 
points  upon  the  ellipse  may  be  easily,  located  and  a  smooth  curve 
drawn  through  them. 

If  the  abscissa  and  ordinate  of  any  point  of  a  curve  are  ex- 
pressed in  terms  of  a  third  variable,  the  pair  of  equations  are 
called  the  parametric  equations  of  the  curve.    Thus, 

X  =  U 
y  =  t  +  l 

are  the  parametric  equations  of  a  certain  straight  line.    Its 
ordinary  equation 

y  =  ix  +  l 

can  be  found  by  eliminating  the  parameter  t. 

Exercises 

1.  Write  the  equation  of  the  ellipse  formed  by  diminishing  the 
lengths  of  all  ordinates  oix'  +  y^  =4  by  one-half  of  their  length. 

2.  Write  the  equation  of  the  ellipse  formed  by  diminishing  the' 
lengths  of  all  ordinates  oi  x'  +  y'  =4  by  one-third  of  their  lengths. 

3.  Write  the  equation  of  the  ellipse  formed  by  lengthening  all 
ordinates  of  the  circle  x'  +  y^  =  16  by  one-third  of  their  length. 

4.  Write  the  equation  of  the  ellipse  formed  by  lengthening  all 
abscissas  of  the  circle  x'  +  ^'  =  1  by  one-fourth  of  their  length. 


§84]  THE  ELLIPSE  AND  HYPERBOLA  157 

6.  Write  the  equation  of  the  ellipse  whose  semi-axes  are  4  and  3. 

6.  Construct  accurately  an  ellipse  whose  semi-axes  are  3  inches  and 
2  inches. 

7.  Construct  accurately  an  ellipse  whose  parametric  equations  are 
x  =  3  cos  e,  and  y  =  2sm  e. 

8.  Write  the  parametric  equations  of  an  ellipse  whose  semi-axes 
are  6  and  10. 

9.  Draw  a  curve  whose  parametric  equations  are  x  =  cos  6,  and 
y  =  sin  e. 

10.  Find  the  major  and  minor  axes  for  the  following : 

(c)  4x'  +  25y'  =  100  (d)  25a;2  +  4y'  =  100 

11.  Find  the  axes  of  the  ellipse  k'x^  +  h'y^  =  hV. 

12.  Write  the  equation  of  the  ellipse  whose  major  and  minor  axes 
are  10  and  6,  respectively. 

13.  Find  the  axes  of  the  elUpse  whose  equation  is 

2/  =  ±  i  V36  —  a;^     [Note  that  o  must  be  6.] 

14.  Write  the  parametric  equations  of  the  ellipse 

y  =  +  IVSl  -x2.     [a  must  be  9;  b  =  f  X  9  =  6.] 
16.  Discuss  the  curve 

a;  =  ±  I  v'4  -  2/2. 

16.  Discuss    the    following    curves    by    comparing    them    with 
a;2  +  2/"  =  1- 

4x2    -1-2/2  =  1 
\x^  H-    2/2  =  1. 

17.  Write  the  Cartesian  equation  of  the  curves  whose  parametric 
equations  are: 

.  ,     Fa:  =  2  cos  9  , .     Tx  =  6  cos  6  i  \    \^  ~  V^^  cos  9 

^°''    ly  =     sin  e  ^  '    ly  =2  sin  S  ^'^'    ly  =  V2  sin  B. 

18.  What  locus  is  represented  by  the  parametric  equations 

X  =  2t  +  1 
2/  =  3«  -f-  5? 

19.  What  curve  is  represented  by  the  parametric  equations 

X  =  2  -H  6  cos  e 
and  2/  =  5  -1-  2  sin  e? 


158        ELEMENTARY  MATHEMATICAL  ANALYSIS 


20.    Show  that  the  curve 

X  =  3  +  3  cos  e 
2/  =  2  +  2  sin  9 
is  tangent  to  the  co6rdinate  axes. 


Rg.  83.— a  mechanical  cons. ru.tion  of  the  ellipse.     See  Exercise  23. 

21.  The  circle  x'  +  y^  =  36  is  picjocted  upon  a  plane.  Find  an 
equation  of  the  projection  if  the  angle  between  the  plane  and  the 
plane  of  the  circle  is  30°. 

22.  A  right  circular  cylinder  is  cut  by  a  plane  making  an  angle  of 
60°  with  the  axis  of  the  cylinder.  Find  an  equation  of  the  curve  of 
intersection,  if  the  radius  of  the  cylinder  is  6  units. 


Fig.  84. — Theory  of  the  common  "ellipsograph"  or  elUptic  trammel. 
See  Exercise  24. 

23.  The  line  AB,  Fig.  83,  whose  length  is  (a  +  6)  moves  in  such  a 
way  that  the  ends  A  and  B  always  lie  on  the  X-  and  K-axes,  respect- 
ively.    Show  that  the  point  P  describes  an  ellipse. 

24.  The- edge  of  a  straight  ruler,  NMP,  Fig.  84,  is  marked  so  that 


§84] 


THE  ELLIPSE  AND  HYPERBOLA 


159 


PM  =  b  and  PN  =  a.  It  is  moved  keeping  M  and  N  always  on 
AA'  and  BB',  respectively.  Show  that  P  describes  an  ellipse.  The 
elliptic  "trammel"  or  "ellipsograph"  is  constructed  on  this  principle 
by  use  of  adjustable  pins  on  PMN  and  grooves  on  AA'  and  BB'. 

25.  Draw  a  semicircle  of  radius  a  about  the  center  C,  Fig.  85,  and 
produce  a  radius  to  0  such  that  CTO  =  a  +  6.  From  C  draw  any 
number  of  lines  to  the  tangent  to  the  circle  at  T.  From  0  draw  hnes 
meeting  the  tangent  at  the  same  points  of  TN.  At  the  points  where 
the  lines  from  C  cut  the  semicircle,  draw  parallels  to  CT.     Show  that 


Fig.  85. — A  graphical  construction  of  an  eUipse.     See  Exercise  25. 


the  points  of  meeting  of  the  latter  with  the  lines  radiating  from  O 
determine  points  on  an  ellipse,  with  center  at  0  and  semi-axes  equal 
to  a  and  b. 

Hint  ;  OD  =  SM  =  a  cos  9.    From  the  triangles  OPD  and  ONT 
OD  _PD 
TN        b 
From  the  triangles  CNT  and  CMS 

SM{=OD)  _  SC{==a  sine) 
TN         ~  a  ' 


160        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§85 

Hence 

PD  =b  sin  e. 

86.  Origin  at  a  Vertex.  Equation  (4)  §80,  equation  (2)  §82,  and 
equations  (1)  and  (2)  §84  are  the  most  useful  forms  of  the  equa- 
tions of  the  ellipse.  It  is  obvious  that  the  ellipse  may  be 
translated  by  the  methods  already  explained  to  any  position  in 
the  plane.  The  ellipse  with  center  at  (A,  k)  and  its  axes  parallel 
to  the  coordinate  axes  has  the  equation 

{x  -  hy      {y  -  kr      .  ,.s 

a2       -r       52       -  ^'  ^^) 

Of  special  importance  is  the  equation  of  the  ellipse  when  the  origin 
is  taken  at  the  left-hand  vertex.  This  form  is  best  obtained  from 
equation  (2),  §82,  by  translating  the  curve  the  distance  a  in  the 
X  direction.    Thus, 

y=±  -Vo»  -  (x  -  ay, 

u 

or 

y^  =  —  X -„  x^, 

a  a^     ' 

or,  letting  21  stand  for  the  coeflBcient  of  x, 

2/2  =  2lx  -~^x^  =  2lxil  -  x/2a).  . (2) 

For  small  values  of  x,  x/2a  is  very  small  as  compared  with  1  and 
the  ellipse  nearly  coincides  with  the  parabola  y^  =  2lx. 

86.  Theorem.  Any  equation  of  the  second  degree,  lacking  the 
term  xy  and  having  the  terms  containing  x^  and  y'  both  present  and 
wUh  coefficients  0/  like  signs,  represents  an  ellipse  with  axes 
parallel  to  the  coordinate  axes.  This  is  readily  shown  by  putting 
the  equation 

ax^  +  hy'  +  2gx  +  2fy  +  c  =  0  (1) 

in  the  form  (1)  of  the  preceding  section.  The  procedure  is  as 
follows: 

a{x'  +  2^^x)  +h[y'  +  2f^y)^-c.  (2) 

a(x^  +  2lx  +  Q    +&(2/^  +  2(.  +  |)  =  f  +  f-c.     (3) 


§87]  THE  ELLIPSE  AND  HYPERBOLA  161 

Let  M  stand  for  the  expression  in  the  right-hand  member  of  (3) ; 
then  we  get 

a  b 

This  shows  that  (1)  is  an  ellipse  whose  center  is  at  the  point 
and  which  is  constructed  from  the  circles  whose  cen- 


\     a'       hi 


ters  are  at  the  same  point  and  whose  radii  are  the  square  roots  of 
the  denominators  in  (4).  The  major  axis  is  parallel  to  OX  or 
OY  according  as  a  is  less  or  greater  than  6.  The  case  when  the 
locus  is  not  real  should  be  noted.     Compare  §43. 

iLLtrsTRATiON:  Find  the  center  and  axes  of  the  ellipse 

a;2    _|-   ^yi    _|_    6j;    _    8?/     =    23. 

Write  the  equation  in  the  form 

x'^  +  Qx  +  4j/2  -Sy  =  23. 

Complete  the  squares 

a"  +  6x  +  9  +  4y2  -  8?/  +  4  =  36. 

Rewriting  (a  +  3)'  +  4(!/  -  l)'  =  36. 

(x  +  3)'   ,    {y  -  ly  ^ 
36       "^        9 

This  is  seen  to  be  an  ellipse  whose  center  is  at  the  point  (—3,  1)  and 
whose  semi-axes  are  6  and  3. 

87.  Limiting  Lines   of  an  Ellipse.    It  is  obvious  from   the 
equation 

y  =  +-^/o''-x^ 

that  a;  =  a  and  x  =.  —  a  are  limiting  lines  beyond  which  the  curve 
cannot  extend;  that  is,  x  cannot  exceed  o  in  numerical  value 
without  y  becoming  imaginary.  The  same  test  may  be  applied 
to  equations  of  the  form 

a;2  +  4x  -h  92/2  -  6?/  -(-  4  =  0. 


162        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§87 
Solving  for  y  in  terms  of  x 


3y  =  l  +  Vl-{x  +  2)2. 

The  values  of  y  become  imaginary  when 

(x  +  2)2>I, 
or 

a:  +  2>+lor<-l, 
or 

x>  -  1  or  <  -  3. 

These,  then,  are  the  limiting  lines  in  the  x  direction.  Finding 
the  limiting  lines  in  the  y  direction  in  the  same  way,  the  rectangle 
within  which  the  ellipse  must  lie  is  determined. 

In  cases .  like  the  above  the  actual  process  of  finding  the  limiUng 
lines  and  the  location  of  the  center  of  the  ellipse  is  best  carried  out 
by  the  method  of  §86. 

Illustration:  Find  the  coordinates  of  the  center,  the  length  of  the 
axes,  and  the  equation  of  the  limiting  lines  of 

x'  +  ix  +  Qy^  -  Qy  =  4. 

Completing  the  squares, 

(X  +  2y  +  9(2/  -  \Y  =  9, 
or 

(X  +  2)'       (y  -  \y  _ 
g—  +         J  1- 

The  center  of  the  ellipse  is  at  the  point  ( —  2,  |),  its  semi-axes  are  3  and 

1.     It  may  be  constructed  by  translating  the  ellipse  -„  +  y  =  1,  two 

units  to  the  left  and  \  unit  up.  Hence  the  limiting  lines  are 
a;  =  +  3  —  2  and  v  =   +  1  +  3,  or  x  =  1,  x  =  —  5,  ?/  =  f ,  and 

y--l 

Exercises 

Find  the  lengths  of  the  semi-axes,  the  coordinates  of  the  center,  and 
the  equations  of  the  limiting  lines  for  the  seven  following  loci  and 
translate  the  curves  so  that  the  terms  in  x  and  y  disappear,  by  the 
method  of  §87. 

1.  x^  -  6x  -I-  42/2  +  82/  =  5. 

2.  2/2  -  8i/  +  4x2  -h  6  =  0. 


§87]  THE  ELLIPSE  AND  HYPERBOLA  163 

8.  12x«  -  48a;  +  3y'  +  6y  -  13. 

4.  x^  +  %2  -  12a;  +  6y  =■  12. 

5.  4a;«  +  y'  -12x  +  12  2/  -  2  =  0. 

6.  x'  +2y'  -  X  -  V2y  =  1/2. 

7.  Show  that  a;^  —  4a;  +  4^^  ^  82/  +  4  =  0  is  an  ellipse. 

8.  Show  that  x'  +  4a;  +  Qy'  —  6y  =  0  passes  through  the  origin. 

9.  Discuss  the  curves : 

,.,£■+!!  =  ,  wM:-' 

10.  Compare  the  following  parabolas  with  the  standard  parabola 
y  =  a;2  by  means  of  the  appropriate  Theorems  on  Loci: 

k 
(a)  y  =  2px2  (c)  j/  =  —  x^ 

(h)  y 2pa;2  (d)  y 2px^  +  6. 

What  are  the  roots  of  the  last  function? 

11.  Write  the  symmetrical  equation  of  the  ellipse  if  its  parametric 
equations  are: 

X  =  (3/2)  cos  e 

y  =  (2/3)  sin  e. 

12.  Discuss  the  curve  y^  =  (18/5)x  -  (9/25)x2. 

13.  Find  the  center  of  the  curve  y^  =  2x  (6  —  x). 

14  Write  the  parametric  equations  for  the  following :_ 

(a)  x^  +  3y'  =4;  (6)  2x^  +  5y^  =  6;  (c)  5x=  +  y^  =  7. 

15.  Write  the  parametric  equations  for 

x'  +2x  +  4;/2  -  l&y  +  13  =  0. 
Hint  ;  The  equation  may  be  put  in  the  form 

(x  +  ir   ,    (y  -  2)' 

Since  x   =   2  cos  6  and  y  =  am.  9  are  parametric   equations  for 

x^        v'^ 

-7   +Y  =  1,  x=2cosfl  —  1  and  2/  =  sin  9   +   2  are  parametric 

equations  of  the  given  ellipse. 

16.  Write  parametric  equations  for  the  following :  (a)  x'  —  2x  + 
9yi  -  6x  =  0;  (6)  4x2  +  4x  +  j/2  -  2?/  =  5.  (c)  3;2  _  4^  ^  ^2  +  gj/  =  3. 


164        ELEMENTARY  MATHEMATICAL  ANALYSIS 


88.  The  Rectangular  Hyperbola.  In  §34  the  graph  of  xy  =■  k, 
where  fc  is  a  constant,  was  called  a  rectangular,  or  equilateral, 
hyperbola.  It  was  observed  that  the  X-  and  F-axes  are  asymp- 
totes of  the  curve.  We  shall  now  find  the  equation  of  the  equi- 
lateral hyperbola  when  rotated  about  the  origin  through  an 
angle  of  (  —  45°).  For  convenience  let  k  be  represented  by  jo^. 
Since  this  is  a  positive  number,  the  curve  will  appear  in  the  first 
and  third  quadrants,  as  shown  by   the   curve  RPS,   Fig.  86. 


Fig.  86. — The  rectangular  hyperbolas  2  xy  —  a'  and  x'  —  y'  =  a'. 

Let  P  be  any  point  on  the  original  curve 

2xy  =  aK  (1) 

Let  P'  be  this  poiat  after  rotation.    Let  OD'  =  x  and  let  B'P'  =  y. 
OB  =  EP  =  E'P'  =  D'K  -  D'H  =  OD'  cos  45°  -  D'P  cos  45° 

=  iV2  {X  -  y).  (2) 

DP  =  OE  =  O'E'  =  OK  +  KE'  =  OD'  cos  45°  +  D'P  cos  45° 

^W2(.x  +  y).  ,  (3) 

But  OD  is  x  and  DP  is  y  in  equation  (1).     Substituting  then 


§89]  THE  ELLIPSE  AND  HYPERBOLA  165 

l\/2(,x  —  y)  and  i\/2{x  +  y)  for  x  and  y,  respectively,  in  equa- 
tion (1),  we  obtain 

x2  -  y2  =  a2,  (4) 

the  equation  of  the  curve  R'P'S',  or  the  equation  of  the  rectangu- 
lar hyperbola  2xy  =  a^  after  it  has  been  rotated  (—  45°)  about 
the  origin. 

The  equation  of  the  asymptotes  of  the  curve  x'^  —  y''  =  a'' 
are  y  =  -\-  x  and  y  =  —  x. 

The  curve  for  xy  =  k  is  sometimes  called  the  equilateral 
hyperbola  referred  to  its  asymptotes  as  axes. 

89.  Parametric  Equations.  The  parametric  equations  of  the 
rectangular  hyperbola  x^  —  y^  =  a^  are 

X  =  a  sec  8,  (1) 

and 

y  =  a  tan  e.  (2) 

For,  dividing  (1)   and   (2)   by  a,  squaring,  and  subtracting, 
-,  -  ^  =  sec"  e  -  tan^  9  =  1, 
which  is  the  same  as  equation  (4)  above. 


Exercises 

1.  Find  the  equations  of  the  following  curves  after  rotation  about 
the  origin  through  an  angle  of  —  45°. 

(o)  2xy  =  1;  (6)  xy  =  1;  (c)  xy  =  4;  (c)  xy  =  f;  (d)  xy  =  3; 
(e)  xy  -  2  =  0. 

2.  Show  that  y^  —  x^  =  a^  is  the  equation  of  the  curve  2xy  =  a' 
after  rotation  about  the  origin  through  an  angle  of  +  45°. 

3.  Show  that  x^  —  y^  =  a^  is  the  equation  of  the  curve  2xy  =  —  o' 
after  rotation  about  the  origin  through  an  angle  of  -|-  45°. 

4.  Find  the  equations  of  the  curves  given  in  exercise  1  after  rota- 
tion about  the  origin  through  an  angle  of  -|-  45°. 

6.  Find  the  equations  of  the  asymptotes  for  x*  —  y^  —  2x  +  4y  =  7. 
Hint:  Completing  the  squares 

{X  -  1)»  -  (2/  -  2)2  =  4 


166        ELEMENTARY  MATHEMATICAL  ANALYSIS 

Since  the  assnnptotes  for  i'  —  ^'  —  4  are  y  ••  ±  x,  the  asymp- 
totes for 

(X  -  1)2  -  (V  -  2)2  =  4  are  2/  -  2  =  ±  (a;  -  1), 
or 

y  =  X  +  1  and  y  +x  =  3. 

6.  Find  the  equations  of  the  asymptotes  and  sketch  the  curves  for 

(a)  x'  -  y'  +2x  +  ^y  =  4; 

(6)  2x^  -  22/2  +  4x  -  81/  =  0. 

90.  The  Hyperbola  of  Semi-axes  a  and  b.  The  ellipse  was 
defined  as  the  curve  produced  by  lengthening  or  shortening  all 
ordinates  of  the  circle  x^  +  y'  =  a',  an  amount  proportional 
to  their  lengths.  Attention  has  been  called  to  the  fact  that  such 
a  curve  results  also  from  the  orthographic  projection  of  the  circle, 
or  from  taking  the  section  of  a  right  circular  cylinder  by  a  plane. 

The  parametric  equations  of  the  circle  are 

X  =  a  cos  6, 
and 

y  =  a  sin  d; 

and  the  parametric  equations  of  the  ellipse  derived  from  this 
circle  as  described  above  are 

X  =  a  cos  d, 
and 

y  =  b  sin  6. 

Let  us  define  the  hyperbola  as  the  curve  obtained  from  the 
equilateral  hyperbola,  x^  —  y^  =  a^,  by  shortening  or  lengthening 
all  ordinates  by  an  amount  proportional  to  their  lengths.  Its 
equation  is  then  obtained  by  replacing  y  in 

x'^  —  y^  =  o'  (1) 

by  my.     Hence  we  have  for  the  equation  of  the  hyperbola 

x^  —  {myy  =  a^. 

a^ 
By  dividing  by  a^  and  replacing  — ^  by  h'^,  we  obtam 


m 
a"      b' 


.-L--^,  (2) 


§91]  THE  ELLIPSE  AND  HYPERBOLA  167 

the  symmetrical  form  of  the  equation  of  the  hyperbola.  It  is 
easily  shown  that 

X  =  a  sec  d,  (3) 

and 

y  =  b  tan  e.  (4) 

are  parametric  equations  of  the  hyperbola  whose  Cartesian  equa- 
tion is  given  by  (2).     For  from  (3)  and  (4)  we  obtain 

-' -|-,  =  sec^e  -  tan^e  =  1. 

Note  that  the  parametric  equations  of  the  equilateral  hyper- 
bola, X  =  asecd  and  y  —  a  tan  6  bear  the  same  relation  to  the 
parametric  equations  of  the  hyperbola,  that  the  parametric 
equations  of  the  circle  bear  to  the  parametric  equations  of  the 
ellipse. 

It  is  seen  that  if  the  ordinates  of  the  asymptotes  to  the  equilateral 
hyperbola  are  affected  in  the  same  way  as  the  ordinates  of  the 
curve  itself,  i.e.,  if  the  asymptotes  are  considered  as  part  of  the 
locus  transformed,  they  are  still  the  asymptotes  to  the  hyperbola 
after  the  transformation. 

The  equations  of  the  asymptotes  of  the  equilateral  hyperbola  are 

y  ==  ±  X. 

By  the  transformation  they  become 

my  =  +  X, 

a 
or,  smce  ™  ~  ft' 

y=±^x,  (5) 

which  are  the  equations  of  the  asymptotes  of 
x^       7/^  _ 

;^2  ~  p  -  1- 

91.  Construction  of  the  Hyperbola.  To  construct  the  hypsr- 
bola  draw  two  concentric  circles  of  radii  OA  =  a  and  OB  =  6, 
as  in  Fig.  87.     Divide  each  circumference,  by  means  of  bow 


168        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§91 

dividers,  into  the  same  number  of  convenient  intervals.  Lay 
ofif,  on  XOX',  distances  equal  to  a  sec  5  by  drawing  tangents  at 
the  points  of  division  on  the  circumference  of  the  a-circle;  also  lay 
off  distances  equal  to  6  tan  6  on  the  vertical  tangent  to  the  6- 
circle  by  prolonging  the  radii  of  the  circle  through  the  points  of 
division  of  the  circumference.  Draw  horizontal  and  vertical 
lines  through  the  points  of  division  of  MN  and  XX',  respectively, 
dividing  the  plane  into  a  large  number  of  rectangles. 


J 

i 

r          1 

—Is/ 

N 
T 

G 

\x 

\, 

y 

yi 

\ 

Y 

^ 

1  /  /y^ 

V^ 

\ 

1 

t  - 

^ 

M 

\ 

\A.' 

w 

A 

D 

/ 

[ 

/ 

\ 

/ 

\ 

■A 

\ 

y. 

y 

f 

bVV' 

\ 

"vV 

Y 

0 

^; 

Fig.  87.— The  hyperbola  xP-ja?-  -  ^fjV  =  1. 


The  point  of  intersection  of  the  vertical  and  horizontal  line 
corresponding  to  the  same  value  of  5  is  a  point  on  the  hyperbola. 
The  curve  may  be  drawn  by  starting  from  the  points  A  and  A' 
and  sketching  the  diagonals  of  successive  rectangles. 

In  the  above  construction,  there  is  no  reason  why  the  diameter 
of  the  6-circle  may  not  be  greater  than  that  of  the  o-circle. 

The  line  A  A'  =  2a  is  called  the  transverse  axis,  the  line  BB'  = 
26  is  called  the  conjugate  axis,  the  points  A  and  A'  are  called  the 
vertices,  and  the  point  0  is  called  the  center  of  the  hyperbola. 


§91]  THE  ELLIPSE  AND  HYPERBOLA  169 

Solving  the  equation  (2)  §90,  for  y,  the  equation  of  the  hyperbola 
may  be  written  in  the  useful  form 

y  =  +  ~  Vx^^T^.  (1) 

Compare  this  equation  with  the  equation  of  the  ellipse,  (2)  §82. 
It  is  easy  to  show  that  the  vertical  distance  PG,  Fig.  87,  of  any 
point  of  the  curve  from  the  asymptote  G'G  can  be  made  as  small 
as  we  please  by  moving  P  outward  on  the  curve  away  from  0. 
Write  the  equation  of  the  hyperbola  in  the  form 

2/1  =  V^^^^,  (2) 

and  the  equation  of  the  asymptote  GG'  in  the  form 

2/2  =  ^s.  (3) 


Then 


PG?  =  2/2  -  2/1  =  ^  (x  -  Va;^  -  a-')  (4) 


Multiply  both  numerator  and  denominator  in  equation  (4)  by 
X  +  Vx^  -  a^. 

PC  —  ~ =^ (5) 

0-X  +  Vx^  -  ffl^ 

Now,  as  X  increases  in  value  without  limit  the  right  side  of  (5) 
approaches  zero.     Whence 

PG  =  Oa,sx  =  a, 


Exercises 

1.  Write  the  symmetrical  equation  of  the  hyperbola  from  the 
parametric  equations  x  =  5  sec  6,  y  =  3  tan  8. 

2.  Find  the  Cartesian  equation  of  the  hyperbola  from  the  relations 
X  =  7  sec  e,  2/  =  10  tan  8.  Note  that  the  graphical  construction  of 
the  hyperbola  holds  if  6  >  u. 

3.  What  curve  is  represented  by  the  equation 

25  16 


170        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§92 


4.  What  curve  is  represented  by  the  equation  y  =  ^Vs'  —  o'? 

5.  Write  the   equation  of  a  hyperbola   having  the   asymptoteB 
y  =  ±.  (3/4)  X,  and  transverse  axis  =  24. 

6.  Show  that  the  curves 


x^  +  6i 


4?/  +  4  =  0 


and 


(.X  +  3)2  -{y  +  2Y  =  l 
are  the  same,  and  show  that  each  is  a  hyperbola. 


Fig.  88. — Conjugate  hyperbolas. 

7.  What  curve  is  represented  by  the  equations 

X  =  h  +  as&a  e 
2/  =  A;  +  5  tan  e? 

8.  Discuss  the  curve  x'  —  8x  —  2y'  —  12y  =  6 . 

92.  Conjugate  Hyperbolas.    Let  GAJ'JA'G',  Fig.  88,   be  the 
hyperbola  whose  equation  is 


6^ 


1. 


(1) 


Its  transverse  axis  is  A  A'  =  2a  and  its  conjugate  axis  is  BB'  =  26. 


*92] 


THE  ELLIPSE  AND  HYPERBOLA 


171 


Its  asymptotes  O'G  and  J' J  are  the  diagonals,  produced,  of  the 
rectangle  constructed  upon  A' A  and  B'B  as  sides. 

The  hyperbola  GBJG'B'J',  having  B'B  as  transverse  axis  and 
A  A'  as  conjugate  axis  and  G'G  and  J' J  as  asymptotes  is  called  the 
conjugate  of  GAJ'JA'G'. 

If  Y'OY  were  the  Z-axis  and  if  X'OX  were  the  Y-axis  the  equa- 
tion   of    the    hyperbola 
GBJG'B'J'  would  be 


0= 


1 


(2) 


By  the  above  supposition  we 
have  interchanged  x  and  y. 
Hence,  to  get  the  true  equa- 
tion we  must  interchange  x 
and  y  in  equation  (2). 
Therefore  the  equation  of 
the  hyperbola  conjugate  to 
x^/a^  —  2/Y62  =  1  is 


r 


(3) 


Fig.  89  shows  a  family  of 
pairs  of  conjugate  hyper- 
bolas. 


Fig.  89. — A  family  of  conjugate 
pairs  of  hyperbolas  with  common 
asymptotes.  (An  interference  pat- 
tern made  from  a  glass  plate  under 
compression.  From  R.  Strauble, 
"TJeber  die  Elsticitats-zahlen  una 
moduln  des  Glases."  Wied.  Ann. 
Bd.  68,  1899,  p.  381.) 


Exercises 

1.  Sketch  on  the  same  pair  of  axes  the  four  following  hyperbolas  and 
their  asymptotes: 


a) 

a;2  -  ^2  = 

25 

(3) 

x' 
25  ■ 

-1- 

(2) 

a;2  _  2,8  =  _  25 
Find  the  axes  of  the  hyperbola 

(4) 

i,y  -- 

X2 

25  ■ 

=  ± 

2/'_ 
9 

1. 

2. 

iVx«- 

64. 

3. 

Sketch  the 

curves: 

a;2 
4 

9 

=  1 

and 

4  ~ 

■'i- 

-       — 

1 

172        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§92 


4.  Sketch  the  curves: 


and 


16 


r 

16 

t 

9 


=  1. 


6.  Write  the  equation  of  the  hyperbola  conjugate  to 

2/  =  ±  f  y/x'  -  64. 
6.  Compare  the  graphs  of: 


2/  =  +  f  Vx^ 

3 


7.  Show  that  3x^ 


Fig.  90. — Diagram  for  Exercise  13. 


64 

2/  =  +  f  Vx'  -  16 

y  =  ±  i  Vx^  -  4 

2/  =  ±  i  Vx^  -  1 

y  =  ±1  Va;'  -  1/16 

2/  =  +  f  Vx^  -  0 

42/'  —  7a;  +  52/  +  2  =  0  is  a  hyperbola.  Find 
the  position  of  the  center  and 
of  the  vertices.  The  vertices 
locate  the  so-called  "limiting 
Unes"  of  the  hyperbola.  Write 
the  equations  of  the  asymptotes. 

8.  Show  that  a;'  -  4a;,  -  ^y^ 
+  42/  =  4  is  a  hyperbola.  Find 
the  coordinates  of  its  center, 
the  equations  of  its  asymp- 
totes, and  the  equations  of  its 
limiting  hnes. 

9.  Discuss  the  graphs: 

x^  -y^  =  1 
and 

2/2  -x 


■>■  =  1. 
=  2,  and  find  the 


10.  Discuss  the  graph  16a;2  —  y'^  —  40a;  — 
hmiting  lines. 

11.  Write  the  equation  conjugate  to 

^  _^  -  1 
4       16 

12.  Write  the  equation  conjugate  to 

a;'  -  2a!  -  2/'  -  62/  =  24. 

13.  A  difficult  problem :  Prove  that  if  a  circular  cylinder  be  cut  by 


§92]  THE  ELLIPSE  AND  HYPERBOLA  173 

a  plane  at  an  angle  of  45°  to  the  axis  of  the  cylinder,  and  if  then  the 
surface  of  the  cylinder  be  unrolled  into  a  flat  surface,  the  curved 
boundary  of  the  surface  is  a  sinusoid.  Thus  if  a  stove  pipe  be  cut 
at  an  angle  of  45°  to  its  axis,  and  if  then  the  sheet  metal  be  unrolled 
into  a  flat  sheet,  the  bounding  curve  is  a  sinusoid. 

In  Fig.  90  only  one  quarter  of  the  cylinder  is  shown.  If  P  be  any 
point  on  the  section  of  the  cylinder  made  by  the  cutting  plane,  and  if 
the  length  of  the  arc  AD  be  called  e  and  the  distance  DP  be  called  y, 
the  problem  is  to  show  that  y  =  sin  8,  provided  the  radius  of  the  cylin- 
der be  called  1.  If  the  angle  of  cutting  be  different  from  45°,  the 
equation  of  the  curve  is  of  the  form  y  =  bain  6,  where  b  =  tan  BOC. 


F{x,  y)  =  xy  sia  -  (2) 


CHAPTER  VI 
SINGLE  AND  SIMULTANEOUS  EQUATIONS 

93.  Notation  of  Functions.  It  has  been  pointed  out  that  the 
symbols  f(,x),  F{x),  4>(,x),  ^{x),  etc.,  are  used  to  denote  functions  of 
X.  Likewise  the  symbols  f{x,  y),  F(x,  y),  <f>(x,  y),^ix,  y),  etc.,  are 
used  to  represent  functions  of  two  arguments  x  and  y.  For 
example,  f{x,  y)  in  a  particular  problem  may  be  used  to  stand 

xy 
for  the  function     /  ,  ,     „■ .    We  may  indicate  this  fact  by  writing 
V  a;2  +y^ 

Again  we  may  abbreviate  the  function  of  x  and  y,  xy  sin  ->  by 
the  symbol  F(x,  y).    This  abbreviation  can  be  indicated  by  writing 

The  equation 

F{x,  2/)  =  0  (3) 

indicates  that  y  is  a  function  of  a;;  y  is  a  function  of  x  expressed 
implicitly.    If  equation  (3)  were  solved  for  y  giving 

V^Kx),  (4) 

J/  is  a  function  of  x  expressed  explicitly.  Equations  (3)  and 
(4)  represent  the  same  functional  relation  between  x  and  y. 
Thus  x^  +  2/2  —  a''  =  0  shows  that  y  is  &  function  of  x  but  the 
functional  relation  is  expressed  implicitly.  If  the  equation  be 
solved  for  y,  giving  y  =  ±  ^a^  —  x"^,  the  same  functional  relation 
between  x  and  y  holds,  but  now  2/  is  an  explicit  function  of  x. 

In  the  same  problem  or  discussion  the  symbols  f{x),  f(y),  f{u), 
or  /(t))  denote  the  same  functional  form  although  the  ftrguments 
may  differ.    If 

^(")  =  V#+t' 

174 


§94]        SINGLE  AND  SIMULTANEOUS  EQUATIONS        175 
f{v)  means  the  same  function  but  with  every  x  replaced  by  y,  thus 

m  -  ^ 


Again,  if  in  a  particular  problem  or  discussion 

f{x)  =x^  +  2x-l, 
then  fiy)  =  y^  +  2y  -  1, 

/(2)  =  2=  +  2-2  -  1  =  7, 
f(-  1)  =  (-  1)^  +  2(-  1)  -  1  =  -  2, 
/(O)  =  0  +  0  -  1  =  -  1. 

Exercises 

1.  URx)  mx'  +  3x+2,  find/(2/);/(3);/tO);/(-l);and/(-2). 

2.  If /(a;)  ^  s'  +  2x^  +  X,  find  A-1);  /(O);  /(+1J;  f(z);  Al/v); 
and  /(<2). 

3.  liF(e)  =sinfl,  findf(,r/2);Ftx);F(0);F(x/6);F(V3)andF(|j-) 

4.  If*>(fl)  s  tanS,  find¥>(0);¥>(7r/6);«>lir/3);#>(ir'/2);¥'(7r)and*>(|ir). 

5.  If /(I,  2/)  ^  -==^'  find/(2,  1);/  (0,  2);/  fe  «)  and/(m,  n). 

V  a;''  +  y' 
Hint:  To  find  /(2,  1)  replace  x  by  2,  and  ?/  by  1  in  the  given  func- 
tion of  X  and  y. 

94.  A  poljmomial  in  x  of  the  nth  degree  is  defined  as 

aox"  +  aia;""'  +  UiX"'^  +  +  a„_ia;  +  On, 

where  the  symbols,  ao,  ai,  02,  .  .  .,  stand  for  any  real  con- 
stants whatsoever,  positive  or  negative,  integral  or  fractional, 
rational  or  irrational,  and  where  n  is  any  positive  integer.  If 
none,  of  the  coefficients  are  zero  the  number  of  terms  in  a 
polynomial  in  x  of  the  nth  degree  is  (n  +  1). 

In  what  follows  in  this  chapter  /(re)  is  supposed  to  stand  for  a 
polynomial  in  x. 

96.  The  Remainder  Theorem.    Let 

f{x)  =  aax"  +  Oia;"-^  +  ajx"-^  H-       .    .   -|-  On-iX  +  On.   (1) 
Then  /(r)  =aor»  +  aif-^  -|-  a^r''-'^  +  -f  (h-\r  +  a„.      (2) 

By  subtracting  (2)  from  (1), 

/(*)  ~  /('■)  —  "oCa;"  —  r")  +  a\{x''~^  —  r"~^)  -f 

+  o^-i(s  -  r).     (3) 


176        ELEMENTARY  MATHEMATICAL  ANALYSIS 

The  right-band  side  of  this  equation  is  made  up  of  a  series  of  terms 
containing  differences  of  like  powers  of  x  and  r,  and,  hence,  by  the 
well-known  theorem  in  factoring,'  each  binomial  term  is  exactly 
divisible  by  {x  —  r).  The  quotient  of  the  right-hand  side  of  (3) 
by  {x  —  r)  may  be  written  out  at  length,  but  it  is  sufficient  to 
abbreviate  it  by  the  symbol  Q{x)  and  write 
fix)  -fir) 


or 


:/^=QW+;^-  (5) 


Equation  (5)  shows  that  f(r)  is  the  remainder  when  f(x)  is 
divided  by  (a;  —  r).    Thus  we  have  the  Remainder  Theorem : 

If  a  polynomial  in  x  be  divided  by  (a;  —  r),  the  remainder  which 
does  not  contain  x  is  obtained  by  writing,  in  the  given  function,  r  in 
place  of  x.  This  theorem  shows,  for  example,  that  the  remainder 
of  the  division 

(.t'  -  6x2  +  iix  _  6)  -f.  (x  -  4)  is  43  -  6(4)2  +  n(4)  -  6,  or  6; 
also  that  the  remainder  of  the  division 

(x'  -  6x2  +  iia;  _  6)  4-  (x  -I-  1) 
is 

(-  1)3  -  6(-  1)2  -1^  11(-  1)  -■  6  =  -  24. 

The  theorem  enables  one  to  write  the  remainder  without  actually 
performing  the  division. 

Exercises 

Without  performing  the  division  find  the  remainder  of  the  following 
divisions : 

1.  (x2  +  3x  -  2)  -=-  (x  -  1). 

2.  (x'  +  3x2  +  2x  -1)  ^  (x  -  2). 

3.  {x*  +  4x=  +  3x2  _  62;  -  1)  -^  (I  +  i). 

4.  (x'  -  3x2  +  2x  -  1)  -^  (x  -2). 
6.  (x2  +  3x  +  2)  +  (x  -h  1). 

6.  (x2  +  3x  +  2)  -i-  {x  +  2). 

96.  Factor  Theorem.  From  equation  (5)  of  the  preceding 
section,  we  see  that  if  fir)  is  zero,  the  remainder  of  the  division  of 

>  See  Appendix,  Chapter  XV,  p.  159. 


§96]        SINGLE  AND  SIMULTANEOUS  EQUATIONS        177 

fix)  by  (x  —  r)  is  zero,  or /(a;)  is  exactly  divisible  by  (x  —  r),  i.e. 
(x  —  r)  is  a  factor  of  f(x).    Thus  we  have  the  Factor  Theorem: 

If  a  polynomial  in  x  becomes  zero  when  r  is  written  in  the  place  of  x, 
{x  —  r)  is  a  factor  of  the  polynomial.  This  means,  for  example,  that 
if  3  be  substituted  for  x  in  the  function  x'  —  6a;^  +  Ha;  —  6  and 
if  the  result  3^  -  6(3)^  +  11(3)  -  6  is  zero,  then  {x  -  3)  is  a 
factor  of  x^  —  Qx'^  +  lis  —  6. 

The  value  r  of  the  argument  x  that  causes  the  function  to  take 
on  the  value  zero  has  already  been  named  a  root  or  a  zero  of  the 
function.  '  The  factor  theorem  may,  therefore,  be  stated  in  the 
form :  A  polynomial  in  x  is  exactly  divisible  by  (x  —  r)  where  r  is  any 
root  of  the  polynomial. 

The  familiar  method  of  solving  a  quadratic  equation  by 
factoring  is  nothing  but  a  special  case  of  the  present  theorem. 
Thus,  if 

x^  -  5x  +  Q  =  0, 

(x  -  2)(x  -  3)  =  0; 

and  the  roots  are  x  =  2  and  x  =  3.  The  numbers  2  and  3  are 
such  that  when  substituted  in  x^  —  5x  +  6  the  expression  is 
zero;  and  the  factors  of  the  expression  are  x  —  2  and  ic  —  3  by 
the  factor  theorem. 

Exercises 

1.  Tabulating  the  cubic  polynomial /(a;)  =  x'  —  6a;*  -|"  H^  —6,  we 
obtain: 

X        -3        -  2      -  1     -  0    1        1.5       2         2.5        3 4 

fix),   -120,    -60,    -24,    -6,  0,  +0.375,  0,   -  0.375,  0,     6 

What  is  the  remainder  when  the  function  is  divided  by  .r  —  4? 
By  I  +  2?     By  a;  +  3?     By  a;  -  1.5?     By  a:  -  3? 
Name  three  factors  of  the  above  function. 

2.  Find  the  remainder  when  a;*  —  5x^  +  12.1;^  +  4a;  —  8  is  divided 
by  a;  -  2. 

3.  Show  by  the  remainder  theorem  that  x"  +  a"  is  divisible  by 
X  +  a  when  n  is  an  odd  integer,  but  that  the  remainder  is  2o"  when  n 
is  an  even  int^er. 

4.  Without  actual  division,  show  that  x*  —  ix'  —  7x  —  24  is 
divisible  by  a;  —  3. 

6.  Show  that  a*  +  a^  —  ab'  —  ¥  is  divisible  by  o  —  b. 

12 


178        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§97 

6.  Show  that  (x  +  l)Ha;  -  2)  -  4(a;  -  l)(a:  -  5)  +  4  is  divisible 
by  I  -  1. 

7.  Show  that  &x^  -  3x*  -  5a;'  +  5a;'  -  2x  -  3  is  divisible  by 
x  +  1. 

8.  Show  that  (b  -  c){h  +  c)'  +  (c  -  a){c  +  a)'  +  (o  -  b)(a  +  b)' 
is  divisible  by  (5  —  c)(_c  —  a)  {a  —  b). 

Hint:  First  consider  the  function  as  a  polynomial  in  6;  then  as  a 
polynomial  in  c;  and  then  as  a  polynomial  in  a. 

9.  Show  that  (6  -  c)'  +  (c  -  o)'  +  (o  -  6)'  is  divisible  by 
(6—  c){,c  —  a)(fl  —  b). 

97.  An  Equation  with  Given  Roots.  The  factor  theorem  enables 
us  to  build  up  a  polynomial  having  given  roots.  If,  for  example,  1, 
2,  and  3  are  roots,  2;  —  1,  x  —  2,  and  a;  —  3,  are  factors  of  the  poly- 
nomial. Hence  (x  —  l)(x  —  2)  (x  —  3),  or  x'  —  6x'  +  llx  —  6 
is  a  factor  of  the  polynomial.  Introducing  another  factor  k,  which 
does  not  contain  x,  cannot  introduce  another  root,  as  a,  for  k  can- 
not contain  the  factor  (x  —  a). 

For  the  same  reason,  multiplying  the  equation  x'  —  6x^  -)-  llx 
—  6  =  0  by  fc,  when  k  does  not  contain  x,  cannot  introduce  roots, 
or  solutions,  in  the  equation.  On  the  other  hand  if  the  equation 
be  multiplied  by  a  function  of  x,  roots  of  the  equation  may  be 
introduced  or  removed.  For,  clearly,  if  the  multiplier  contains  the 
factor  (x  —  a),  the  root  a  will  be  introduced;  and  if  the  multiplier 
contains  the  factor  (x  —  1)  in  its  denominator,  the  factor  (x  —  1) 
will  be  divided  out  from  both  numerator  and  denominator,  if  it 
is  a  factor  of  the  numerator  and  the  root  1  wiU  be  removed  from 
the  function. 

Exercises 

Build  up  polynomial  equations  having  the  following  numbers  for 
roots: 

1.  1,  3,  and  4.  2.  -1,  2,  and  -3.  3.  0,  2,  and  -1.  4.  1,  0,  0, 
and  2. 

98.  Legitimate  and  Questionable  Transformations.  If  one 
equation  is  derived  from  another  by  an  operation  which  has  no 
effect  one  way  or  another  on  the  solution,  it  is  spoken  of  as  a 
legitimate  transformation ;  if  the  operation  is  of  such  a  nature  that 
it  may  ha.ve  an  effect  upon  the  roots,  it  is  called  a  questionable 


§98]        SINGLE  AND  SIMULTANEOUS  EQUATIONS        179 

transfonnation,  meaning  thereby  that  the  effect  of  the  operation 
requires  examination. 

In  performing  operations  on  the  members  of  equations,  the 
effect  on  the  solution  must  be  noted,  and  proper  allowance 
made  in  the  result.  It  cannot  be  too  strongly  emphasized  that 
the  test  for  any  solution  of  an  equation  is  that  it  satisfy  the  original 
equation.  "No  matter  how  elaborate  or  ingenious  the  process 
by  which  the  solution  has  been  obtained,  if  it  do  not  stand  this 
test  it  is  no  solution;  and,  on  the  other  hand,  no  matter  how  simply 
obtained,  provided  it  do  stand  this  test,  it  is  a  solution."^ 

By  the  principles  or  axioms  of  algebra,  an  equation  remains 
true  if  we  unite  the  same  number  to  both  sides  by  addition  or 
subtraction;  or  if  we  multiply  or  divide  both  members  by  the 
same  number,  not  zero;  or  if  like  powers  or  roots  of  both  members 
be  taken.  But  we  have  indicated  in  the  preceding  section  that 
these  operations  may  affect  the  number  of  roots  of  the  equation. 
This  is  obvious  enoHgh  in  the  case  already  cited.  Sometimes, 
however,  the  operation  that  removes  or  introduces  roots  is  so 
natural  and  its  effect  is  so  disguised  that  the  student  is  apt  not  to 
take  due  account  of  its  effect.    Thus,  the  roots  of 

3(x  -  5)  =  x(x  -  5)  +  x'  -  25  (1) 

are  —  1  and  5,  for  either  of  these  when  substituted  for  x  will 
satisfy  the  equation.  Dividing  the  equation  through  by  a;  —5, 
the  resulting  equation  is 

3  =  a;  +  a;  +  5. 
This  equation  is  not  satisfied  by  a;  =  5.     One  root  has  disappeared 
in  the  transformation.    It  is  easy  to  keep  account  of  this  if  (1) 
be  given  in  the  form 

(a;  -  5)(a;  +  1)  =  0, 
but  the  fact  that  a  factor  has  been  removed  may  be  overlooked 
when  the  equation  is  written  in  the  form  first  given. 

A  very  important  effect  upon  the  roots  of  an  equation  results 
from  squaring  both  members.  The  student  must  always  take 
proper  account  of  the  effect  of  this  common  operation.  To  il- 
lustrate, take  the  equation 

a;  +  5  =  1  -  2a;.  (2) 

'  Chrystftl's  Algebra, 


180        ELEMENTARY  MATHEMATICAL  ANALYSIS 

It  is  satisfied  only  by  the  value  a;  =  —  f .  Now,  by  squaring 
both  sides  of  the  equation,  we  obtain 

a;''  +  lOs  +  25  =  1  -  4x  +  ix', 

which  is  satisfied  by  either  a;  =  6  or  a;  =  —  |.  Here  obviously, 
an  extraneous  solution  has  been  introduced  by  the  operation  of 
squaring  both  members. 

It  is  easy  to  show  that  squaring  both  members  of  an  equation 
is  equivalent  to  multiplying  both  sides  by  the  sum  of  the  left  and 
right  members.    Thus,  let  any  equation  be  represented  by 

L(x)  =  R^x)  (3) 

in  which  L(x)  represents  the  given  function  of  x  that  stands  on 
the  left-hand  side  of  the  equation  and  R{x)  represents  the  given 
function  of  x  that  stands  on  the  right-hand  side  of  the  equation. 
Squaring  both  sides, 

[Lix)]^  =  [B(,x)V. 
Transposing, 

[L(,x)r  -  [R\x)V  =  0, 
factoring, 

[Lix)  +  R(x)]  mx)  -  Rix)]  =  0. 

But  (3)  may  be  written 

L{x)  -  R{x)  =  0. 

Thus,  by  squaring  the  members  of  equation  (3)  the  factor 
L(x)  +  R(x)  has  been  introduced. 

The  sum  of  the  left-  and  right-hand  members  of  (2),  above,  is 
6  —  a;.  Hence,  squaring  both  sides  of  (2)  is  equivalent  to  the 
introduction  of  this  factor,  or  thq  operation  introduces  the  root 
6,  as  already  noted. 

As  another  example,  suppose  that  it  is  required  to  solve 

sin  a  cos  a  =  \  (4) 

for  a  <  90°.     Substituting  for  cos  a,  Equation  (4)  becomes 

sin  aVl  —  sin"  a.  -  \,  (5) 

squaring 

sin^  a(l  —  sin''  a)  =  y^, 


§98]         SINGLE  AND  SIMULTANEOUS  EQUATIONS  181 

completing  the  square 

sin*  a.  —  sin''  a-\-\  =  j-^-. 
Hence, 

sin  a  =  +  Vi  +  i  \/3 

=•  ±  0.9659  or  ±  0.2588. 

Only  the  positive  values  satisfy  (4);  the  negative  values  were 
introduced  in  squaring  (5).  If,  however,  the  restriction  a  <  90° 
be  removed,  so  that  the  radical  in  (5)  must  be  written  with  the  double 
sign,  then  no  new  solutions  are  introduced  by  squaring. 

Among  the  common  operations  that  have  no  effect  on  the  solu- 
tion are  multiplication  or  division  by  known  numbers,  or  addition 
or  subtraction  of  like  terms  to  both  members;  none  of  these  intro- 
duce factors  containing  the  unknown  number.  Taking  the 
square  root  of  both  numbers  is  legitimate  if  the  double  sign  be 
given  to  the  radical.  Clearing  of  fractions  is  legitimate  if  it  be  done 
so  as  not  to  introduce  a  new  factor.  If  the  fractions  are  not  in 
their  lowest  terms,  or  if  the  equation  be  multiplied  through  by  an 
expression  having  more  factors  than  the  least  common  multiple 
of  the  denominators,  new  solutions  may  appear,  for  extra  factors 
are  probably  thereby  introduced.  Hence,  in  clearing  of  fractions, 
the  multiplier  should  be  the  least  common  denominator  and  the 
fractions  should  be  in  their  lowest  terms.  This,  however,  does  not 
constitute  a  sufficient  condition,  therefore  iAe  only  certainty  lies 
in  checking  all  results. 

Exercises 

Suggestions:  It  is  important  to  know  that  any  equation  of  the 
form 

oa;2»  +  bx''  +  c  =  0 

can  be  solved  as  a  quadratic  by  finding  the  two  values  of  a;".  Fre- 
quently equations  of  this  type  appear  in  the  form 

dx'  +  ex~^  =  f. 

Likewise  any  equation  of  the  form 

aj(x)  +  6V7(S)  +  c  =  0 

can  be  solved  as  a  quadratic  by  finding  the  two  values  of  VjCi)  and 


182        ELEMENTARY  MATHEMATICAL  ANALYSIS 

then  solving  the  two  equationa  resulting  from  putting  ■\/f(x)  equal  to 
each  of  them.    One  of  these  usually  gives  extraneous  solutions. 

These  two  tjrpes  occur  in  the  exercises  given  below. 

Since  operations  which  introduce  extraneous  solutions  are  often 
used  in  solving  equations,  the  only  sure  test  for  the  solution  of  any 
equation  is  to  check  the  results  by  substituting  them  in  the  original 
equation. 

Take  account  of  all  questionable  operations  in  solving  the  following 

equations: 

3a;  6,9 

+ 7.-     Note  :  3  la  not  a,  root. 


'  X  -3      a;  +  3a;-3 

2.  {x^  +  5x  +  6)/{x  -  3)  +  4a;  -  7  =  -  15. 

3.  3(a;  -  5){x  -  l){x  -  2)  =  (x  -  5)(x  +  2)(.x  +  3). 

Note  :  Divide  by  (a;  —  5),  but  take  account  of  its  effect. 

4.  x'/a  +  ax  =  x''/h  +  6a;. 

6.  oa;(ca;  —  36)  =  5o(36  —  ex). 

6.  a;'  —  Ji*  =  n  —  a;. 

7.  (a;  -  4)»  +  (a;  -  5)»  =  31[(a;  -  4)^  -  (a;  -  5)'^].  Divide  by 
(a;  -  4)  +  (a;  -  5)  or  2x  -  9. 

a;^  —  3a;  1 

8.  ■  _  . — h  2  +  _  -  =0.  If  the  fractions  be  added,  multi- 
plication is  unnecessary.    There  is  only  one  root. 

9.  X  =  1  -  Va;'  -  7. 

10.  Va:  +  20  -  Va;  -  1  -  3  =  0. 

11.  \/l5/4  +  a;  =  3/2  + -y/g.  

12.  20a;/ V  10a;  -  9  -  VlOx  -  9  =  IS/VlOa;  -9+9. 

13.  — ;= ,  = ;;.     Consider  as  a  proportion  and  take 

y/x-  y/x-Z      ^-^ 

by  composition  and  division. 

14.  a;^_+  5/2  =  (13/4)x>^. 

16.  y/x^  -  2y/x  +  a;  =  0.     Divide  by  y/~x. 

16.  2V'a;2  -5x  +  2  -  x'  +  8x  =  3x  -  6.     Call  a;^  -  5a;  +  2  =  u'. 

17.  4a;2  -  4a;  +  20\/2a;2  -  5a;  +  6  =  6a;  +  66. 

18.  x-^  -  2a;-i  =  8.  22.  8x^  -  Sx'^  =  63. 

19.  x^^  -  5a;^i  +4  =  0.  23.  (x  -  a)"  -  3(x  -  a)-»  =  2. 

20.  110a;-*  +  1  =  21a;-2  24.  2a;^  -  3a;^  +  x  =  0. 

21.  Vx  +  4x-J^  =  5. 

99.  Intersection  of  Loci.  In  §41  it  was  shown  that  the  coordi- 
nate of  the  points  of  intersection  of  two  loci  could  be  found  by 
solving  the  equations  of  the  loci  considered  as  simultaneous 
equations. 


SINGLE  AND  SIMULTANEOUS  EQUATIONS         183 


Let  all  terms  of  an  equation  be  transposed  to  the  left-hand 
member,  rendering  the  right-hand  member  zero.  Let  this  left- 
hand  member  be  abbreviated  by  u.  The  equation  then  takes  the 
form 

M  =  0.  (1) 


In  a  similar  way  let  a  second  equation  be  put  in  the  form 


«  =  0. 


(2) 


Fig. 


91 . — Intersections 
curves. 


Let  the  graphs  for  equations  (I)  and  (2)  be  represented  in  Fig. 
91.    The  coordinates  of  any  point  on  curve  (1)  make  u  equal  to 
zero.    The  coordinates  of  any  point 
on  curve  (2)  make  v  equal  to  zero. 

Consider  the  graph  of 

u  +  kv  =  Q,  (3) 

where  h  is  any  constant.  The  co- 
ordinates of  a  point  of  intersection 
of  the  u  and  v  curve  satisfy  equa- 
tion (3).  For,  these  coordinates 
make  u  zero  and  they  make  v  zero, 
then  they  make  u  +  kv  zero..  Fur- 
ther, the  coordinates  of  a  point  on 
the  u  curve  which  is  not  on  the  v 

curve  do  not  satisfy  equation  (3).  For  these  coordinates  make 
u  zero  but  do  not  make  v  zero,  then  they  do  not  make  u  +  kv 
zero.  Similarly  the  coordinates  of  a  point  on  the  v  curve  which 
is  not  on  the  u  curve  do  not  satisfy  equation  (3).  Hence  the 
graph  of  (3)  passes  through  all  points  of  intersection  of  the 
u  and  V  curves  but  does  not  intersect  these  curves  in  any  other 
points.  Thus  to  find  the  coordinates  of  the  points  of  intersec- 
tion of  the  u  and  v  curve  we  may  solve  (1)  and  (3)  or  (2)  and  (3) 
as  simultaneous  equations. 
The  locus  of  the  equation 

WW  =  0  (4) 

is  the  M  and  v  curves  considered  as  a  single  locus.  For,  the  coordi- 
nates of  a  point  on  the  u  curve  make  u  zero,  then  they  make  uv 
zero.    Similarly  the  coordinates  of  a  point  on  the  v  curve  make 


184        ELEMENTARY  MATHEMATICAL  ANALYSIS 

uv  zero.  The  coordinates  of  a  point  neither  upon  the  u  curve 
nor  upon  the  v  curve  make  neither  u  nor  v  zero,  then  they  cannot 
make  uv  zero.  Hence  the  locus  of  (4)  consists  of  all  points  on 
the  u  and  v  curve  but  of  no  other  points. 

To  find  the  points  of  intersection  of  the  circle  x^  +  y'  =  25  and  the 
straight  line  x  +  y  =  7  yre  solve  the  equations  by  the  usual  method,  as 
follows: 

x^  +  y'  =  25\  (5) 

X   +y    =    7j  (6) 

The  graphs  are  a  circle  and  a  straight  line,  as  shown  in  (1),  Fig.  92. 
Squaring  the  second  equation,  the  system  becomes 

x^+y'>  =  25\  (7) 

X'  +  2xy  +  2,2  =  49  /  (8) 

The  second  equation  represents  the  two  straight  lines  shown  in  (2) 
Fig.  92.  The  effect  of  squaring  has  been  to  introduce  two  extraneous 
solutions  corresponding  to  the  points  Ps  and  Pt.  For,  eCiuation  (8) 
may  be  written  {x  +  y  +  7)ix  +  y  —  7)  =0  while  (6)  from  which  it 
was  derived  is  x  +  y  —  7  =  0. 

Multiplying  (7)  by  2  and  subtracting  (8)  from  it,  the  last  pair  of 
equations  becomes 

x^  -2xy  +  y*  =    l\  (6) 

x'  +  2xy  +  v"  =  49  J  (7) 

which  gives  the  four  straight  lines  of  Fig.  92,  (.4).  Taking  t^e  square 
root  of  each  member,  but  discarding  the  equation  x  +  y  +  7  =  0, 
because  it  corresponds  to  the  extraneous  solutions  introduced  by  the 
questionable  operation,  we  have: 

x-y  =  ±l\  (8) 

(9) 


-2/=  ±1\ 
+  y  =7      / 

By  addition  and  subtraction  we  obtain  the  results: 

(10) 


X  =  S\ 
y=4J 

a;  =4"! 
2/  =  3/ 


(11) 


represented  by  the  intersections  of  the  lines  parallel  to  the  axes  shown 
in  Fig.  92,  (5). 


§99]         SINGLE  AND  SIMULTANEOUS  EQUATIONS         185 

This  is  a  good  illustration  of  the  graphical  changes  that  take  place 
during  the  solution  of  simultaneous  equations  of  the  second  degree. 
The  ordinary  algebraic  solution  consists,  geometrically,  in  the  succes- 
sive replacement  of  loci  by  others  of  an  entirely  different  kind,  but  all 
passing  through  the  points  of  intersection  (as  Pi,  Pa,  Fig.  92)  of  the 
original  loci.     The  final  locii  are  straight  lines  parallel  to  the  axes. 


FiQ.  92. — Graphic  representation  of  the  steps  in  the  solution  of  a 
certain  set  of  simultaneous  equations. 


Exercises 

Find  the  coordinates  of  the  points  of  intersection  of  the  following 
pairs  of  equations;  sketch  curves  representing  all  equations  involved 
in  the  solution: 


1.  xy  =  1 

3x  -  5y  =  2 

2.  X'  +  y'  =  5 

y^  =  4x 

Hint  fob  Ex.  4:  Let  u  =  x^andw 


3.  x'  +  X  =  4j/* 
3j  +  6j/  =  1 

4.  x2  +  2/2  =  9 

a;2   -  yi   =  4 

=  1/2.     Solve  for  u  and  v. 


186        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§100 

Solve  graphically  the  following : 

6.  x'  +  y'  =  25  6.  x'  +  y^  =  25 

X   +y    =2  x^  +y^  +  2x  -6y  +  6  =0 

7.  y  =  x^  +  X  —  I 
xy  =  1. 

100.  Quadratic  Systems.'  Any  linear-quadratic  systena  of 
simultaneous  equations,  such  as 

y  =  mx  +  k 

ax''  +  hy^  +  2hxy  -\- 2gx  +  2fy  +  c  =  0 

can  always  be  solved  analytically;  for  y  may  readily  be  eliminated 
by  substituting  from  the  first  equation  into  the  second.  A 
system  of  two  quadratic  equations  may,  however,  lead,  after 
elimination,  to  an  equation  of  the  third  or  fourth  degree;  and, 
hence,  such  equations  cannot,  in  general,  be  solved  until  the 
solutions  of  the  cubic  and  bi-quadratic  equations  are  known. 

A  single  illustration  will  show  that  an  equation  of  the  fourth 
degree  may  result  from  the  elimination  of  an  unknown  number 
between  two  quadratics.    Thus,  let 

x^  —  y   =  5x 

a;2  +  j/2  =  10. 

From  the  first,  y  =  x^  —  5x.  Substituting  this  value  of  y  in  the 
second  equation,  and  performing  the  indicated  operations,  we 
obtain 

a;4  _  lOx'  +  26s^  -  10  =  0. 

WhUe,  in  general,  a  bi-quadratic  equation  results  from  the 
process  of  elimination  from  two  quadratic  equations,  there  are 
special  cases  of  some  importance  in  which  the  resulting  equation 
is  either  a  quadratic  equation  or  a  higher  equation  in  the  quadratic 
form.     Two  of  these  cases  are: 

(1)  Systems  in  which  the  terms  containing  the  unknown  num- 
bers are  homogeneous;  that  is,  systems  in  which  the  terms  con- 

^  A  large  part  of  the  remainder  of  this  chapter  can  be  omitted  if  the  students 
have  had  a  good  course  in  algebra  in  the  secondary  school. 


§101]       SINGLE  AND  SIMULTANEOUS  EQUATIONS        187 

taining  the  unknown  numbers  are  all  of  the  second  degree  with 
respect  to  the  unknown  numbers,  such,  for  example,  as 

x''  —  2xy   =    5 

3x^  -  lOy^  =  35. 

(2)  Systems  in  which  both  equations  are  symmetrical;  that  is, 
such  that  interchanging  x  and  y  in  every  term  does  not  alter  the 
equations;  for  example 

x'  +  y^  -  X  -  y  =  78 

xy  +  X  +  y  =  39. 

101.  Unknown  Terms  Homogeneous.  The  following  work 
illustrates  the  reasoning  that  will  lead  to  a  solution  when  applied 
to  any  quadratic  system  all  of  whose  terms  containing  x  and  y 
are  of  the  second  degree.    Let  the  system  be 

x^  —  xy  =  2 

2x^  +  2/2  =  9.  (1) 

Divide  each  equation  by  x'^  (or  y'^),  then 

1  -  iy/x)  =  2/x^ 

2  +  (y/xr  =  Vx'.  (2) 

Since  the  left  members  were  homogeneous,  dividing  by  x'  renders 
them  functions  of  the  ratio  (y/x)  alone;  call  this  ratio  m.  Then 
equations  (2)  contain  only  the  unknown  numbers  m  and  x^. 
The  latter  is  readily  eliminated  by  subtraction,  leaving  a  quad- 
ratic for  the  determination  of  m.  When  m  is  known,  substituting 
in  (2)  determines  x,  and  the  relation  y  =  mx  determines  the 
corresponding  values  of  y. 

The  above  illustrates  the  principles  on  which  the  solution  is  based. 
In  practice,  it  is  usual  to  substitute  y  =  mx  at  once,  and  then  eliminate 
x'  by  comparison;  thus,  from  the  substitution  y  =  mxin  (1),  we  obtain 

x'  -  mx^  =  2 

2x'  +  mV  =  9.  (3) 


188        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§101 
Thence, 


Whence, 
or 


a;2  =  2/(1  -  m) 
x^  =  9/(2  +  m'). 

2/(1  -m)  =  9/(2  +  m^), 

2m'  +  9m  =  5. 


\ 

\ 

Y 

\ 

\ 
\ 
\ 

4 

/ 

V 

y// 

\ 

3 

// 

u\ 

2          \^ 

1      y/ 

X'                     /       J 

'                              JC 

-4        -3        -  2  /-I   ^^Z' 

% 

2         3         4 

/ 

// 

A 

-  a  =  -  Vs  Vi" 

// 

•4    \ 
\ 

' 

\ 
\ 

Y 

1 

(4) 
(5) 
(6) 


Fig.  93. — Solutions  of  a  set  of  simultaneous  quadratics  given  graph- 
ically by  the  coordinates  of  the  points  of  intersection  of  an  ellipse  and 
hyperbola. 


Factoring, 

whence. 

Hence, 


(2m  -  l)(m  4-  5)  =  0 

m  =  1/2  or  —  5. 

X  =  +  2  or  +  (l/3)-v/3 
2/  =  +  1  or  +  (5/3)v'3. 


(7) 
(8) 

(9) 


§102]       SINGLE  AND  SIMULTANEOUS  EQUATIONS        189 

These  solutions  should  be  written  as  corresponding  pairs  of  values  as 

follows: 

X  =  2  X.  =  -2  X  =        (1/3)V3         a;  =  -  (l/3)\/3 

y  =  l  v=-l  t/='-(5/3)V3         y=       (5/3)^3 

This  system  can  readily  be  solved  without  the  use  of  the  mx  sub- 
stitution by  merely  solving  the  first  equation  fpr  y  and  substituting 
in  the  second. 

Graphically  (See  Fig.  93),  the  above  problem  is  equivalent  to 
finding  the  intersections  of  the  curves : 

x(x  -  y)  =  2 
(V2x)'  +  y'  =  9 
The  first  is  a  curve  with  the  two  asymptotes  x  =  0  and  x  —  y  =  0. 
That  these  lines  are  asymptotes  is  readily  seen  if  the  equation  be 
2 
put  in  the  form  y  =  x If  a;  is  positive,  y  is  less  than  x,  or  the 

curve  is  below  the  line  y  =  x.     If  x  is  negative  y  is  greater  than  x,  or 

the  curve-is  above  the  line  y  =  x.     As  x  increases  in  numerical  value, 

2 

-  approaches  zero  and  the  curve  approaches  the  line  y  =  x.     As  a; 

approaches  zero,  y  increases  without  limit.  As  a  matter  of  fact,  the 
curve  is  a  hyperbola,  although  proof  that  such  is  the  case  cannot  be 
given  until  the  method  of  rotating  any  curve  about  the  origin  has  been 
explained.  The  second  curve  is  obviously  an  ellipse  generated  from 
a  circle  of  radius  3  by  shortening  the  abscissas  in  the  ratio  ■y/2  : 1.  The 
two  curves  intersect  at  the  points: 

X  =2  -  2  0.557   .  .  .  -0.557   ... 

2/  =  1  -  1  -  2.887   ...  +2.887   ... 

The  auxiliary  lines,  y  =  ^x  and  y  =  —  5x,  made  use  of  in  the  solution 
are  shown  by  the  dotted  lines. 

102.  Symmetrical  Systems.  Simultaneous  quadratics  of  this 
type  are  readily  solved  analytically  by  solving  for  the  values  of 
the  binomials  x  -\-  y  and  x  —  y.  The  ingenuity  of  the  student 
will  usually, show  many  short  cuts  or  special  expedients  adapted 
to  the  particular  problem.  The  following  worked  examples  point 
oat  some  of  the  more  common  artifices  used. 

1.  Solve 

x  +  y  =Q  (1) 

xy  =  5.  (2) 


190        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§102 

Squaring  (1) 

x^  +  2xy  +  y^  =  36.  (3) 

Subtracting  four  times  (2)  from  (3) 

x'-  -  2xy  +  2/=  =  16. 


Whence 

But  from  (1) 

Therefore 

a;  =  5 
2/  =  l 
2.  Solve 

X  -y  = 
X  +y  = 

and 

X2    +  2/2    : 

±  4. 
6. 

=  34 

a;  =  1 
2/  =5. 

(1) 

xy  ■■ 
Adding  two  times  (2)  to  (1) 

=  15 

(2) 

x' 

>  +  2xy  + 

y'  =  64. 

(3) 

Subtracting  two  times  (2)  from  (1) 

X 

Whence,  from  (3)  and  (4) 

2  -  2xy  + 
X  +y  = 

2/2=4- 
±8 

(4) 

Therefore 

X  -  V  = 

±2. 

X  =  5               X  =3 

a;  =  -  5 

'          X  = 

-3 

^  =  3               y  =  5 

y  =  -3 

y  = 

-S 

The  hyperbola  and  circle  j 
by  the  student. 
3.  Solve 

represented  by  (1)  and  (2)  should  be  drawn 
x>  +  y^  =  72                                             (1) 

X   +y    = 
Cubing  (2) 

a;3  4-  3a;2j,  ^  ^xy' 

=    6. 

'  +  y'  =  216. 

(2) 
(3) 

Subtracting  (1)  and  dividing  by  3 

4 

whence,  since 

xy(x  +  y) 
X  +y 

=  48, 
=  6 

(4) 

we  have 

y 

=  8. 

(5) 

§102]       SINGLE  AND  SIMULTANEOUS  EQUATIONS        191 

From  (2)  and  (5)  proceed  as  in  example  1,  and  find 

1  =  4  ,  X  =  2. 

r>  and  , 

y  =  2  2/  =  4 

Otherwise,  divide  (1)  by  (2)  and  proceed  by  the  usual  method. 
4.  Solve 

a;2  +  SI/  =  ^  (a;  +  J/)  (1) 

y-'-'rxy  =  ^-  (x  +  v).  (2) 

Adding  (1)  and  (2) 

{x  +  yy  -  6{x  +  y)  '=  0,  (3) 

whence, 

X  +  2/  =  0  or  6.  (4) 

Now,  because  x  +  y  is  a  factor  of  both  members  of  (1)  and  (2),  the 
original  equations  are  satisfied  by  the  unlimited  number  of  pairs  of 
values  of  x  and  y  whose  sum  is  zero,  namely,  the  coordinates  of  all 
points  on  the  line  x  -\-  y  =  Q. 
Dividing  (1)  by  (2),  we  get 

x/y  =  7/11. 

This,  and  the  line  x  -\-  y  =  &,  from  (4),  give  the  solution: 

y  =  T/Z 

y  =  11/3. 

Graphically,  the  equation  (1)  is  the  two  straight  lines  i; 

{x-7/3){x  +  y)  =0. 

Equation  (2)  is  the  two  straight  lines 

(2/  -  n/3){x+y)  =0. 

These  loci  intersect  in  the  point  (7/3,  11/3)  and  also  intersect  every- 
where on  the  line  x  +  y  =  0. 

Exercises 
1.  Show  that 

3-2   +  J/2    =    25 

X  +  y  =  1 
has  a  solution,  but  that  there  is  no  real  solution  of  the  system 
a;2  +  j/2  =  25 
X  +y  =  U. 


192        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§103 
2.  Do  the  curves 


Do  the  curves 


3.  Solve 


x'  +  y'  =  25 

xy  —  100,  intersect? 

a;2  +  2/2  =  25 

xy  =  12,  intersect? 

(x^  +  y'){x  +  y)  =  272 
x^  +  y^  +  x'+  y  =  42. 


Note  :  Call  x^  +  y^  =  u,  and  x  +  y  =  v. 

4.  Show  that  there  are  four  real  solutions  to 

x^  -\-  y^  -  \2  =  X  -\-y 

xy-\-S  =  2{x  +y). 

5.  Solve  x''  -\- y^  -\-  x  +  y  =  li 

xy  =  6. 

103.  Graphical  Solution  of  the  Cubic  Equation.  The  roots  of  a 
cubic  x'  +  ax'  +  j3x  +  7  =  0  (where  a,  /3,  and  7  are  given  known 
numbers)  may  be  determined  graphically  as  explained  in  §40. 

Another  method  of  solving  the  cubic  equation  graphically 
will  now  be  given.    The  roots  of  the  equation 

x^  +  ax^  +  ^x  +  y  =  0  (1) 

are  the  JST-intercepts  for  the  graph  of 

y  =  x^  +  ax'  +  ^x  +  y.  (2) 

If  we  replace  x  in  equation  (2)  by  (x  —  k),  where  fc  is  a  constant, 
the  equation  (2)  becomes 

y  =  ix-ky+  a(.x  -  ky  +  /3(a;  -  k)  +  y, 
or 

y  =  x^  +  {a-  Zh)x'  +  (j3  -  2ak  +  ^k')x 

-{¥  -  al<!,'+  fik  -y).     (3) 

a 
It  will  be  noticed  that  if  k  is  chosen  equal  to  -5  the  coefficient  of 

x^  vanishes  and  equation  (3)  becomes 

y  =  x^  +  ax  +  b,  (4) 


§103]       SINGLE  AND  SIMULTANEOUS  EQUATIONS         193 


when  a  stands  for  the  coefficient  of  x  and  h  stands  for  the  absolute 
term  of  equation  (3). 
Since  the  graph  for  (4)  differs  from  the  graph  of  (2)  only  in  that 

it  is  translated  -s  units  parallel  to  the  Z-axis,  the  X-intercepts  for 

the  first  graph  are  ^  units  greater  (less  if  a  is  negative)  than  the 
X-intercepts  for  the  second  graph.    Hence,  if  the  roots  of 

x^  +  ax  +  h  (5) 

can  be  found,  these  roots  decreased  by  o  are  the  roots  of  (1). 

Since  an  equation  of  the  form 
(2)  can  always  be  put  in  the  form 
of  equation  (4),  we  shall  only 
consider  cubic  equations  of  the 
form 

x^  +  ax  +  h  =  Q.         (6) 

Consider  the  system  of  curves 

2/  =  x=  (7) 

y  =  —  ax  —  b.  (8) 

Equation  (7)  gives  the  cubic  pj^  9 4.— Construction  for 
parabola,  and  (8)  the  straight  graphical  solution  of  .t'  +  a  x+ 
line.  Fig.  94.  ^  =  "• 

Let  P  be  a  point  of  intersec- 
tion of  the  cubic  and  the  straight  line.    Let  OD  be  the  abscissa 
of  the  point  P.     The  value  of  OD  is  a  root  of  equation  (6).     For 
OD  is  a  value  of  x  for  which  a;'  =  —  ax  —  b,  or  for  which  a;'  + 
ax  +  b  =  0. 

In  drawing  the  graph  of  the  cubic  parabola,  it  is  desirable  to  use,  for 
the  ^-seale,  one-tenth  of  the  unit  used  for  the  x-scale,  so  as  to  bring  a 
greater  range  of  values  for  y  upon  an  ordinary  sheet  of  coordinate 
paper.  The  cubic  parabola  graphed  to  this  scale  is  shown  in  Kg.  95. 
The  diagram  gives  the  solution  of  s'  —  a;  —  1  =  0.  The  graphs 
y  =  x'  and  y  =  x  +  1  aie  seen  to  intersect  at  x  =  1.32.  This,  then, 
should  be  one  root  of  the  cubic  correct  to  two  decimal  places.  The 
line  y  =  X  +  1  cuts  the  cubic  parabola  in  but  one  point,  which  shows 
that  there  is  but  one  real  root  of  the  cubic.     To  obtain  the  imaginary 


194        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§103 


Fia.  95. — Graphic    solution    of    the    cubic    a;^  —  a;  —  1  =  0    and 
s3  _  10  a;  -  10  =  0. 


§103]       SINGLE  AND  SIMULTANEOUS  EQUATIONS        195 

roots,  divide  a;^  —  a;  —  1  by  x  —  1.32.  The  result  of  the  division, 
retaining  but  two  places  of  decimals  in  the  coefficients,  is 

s2  +  1.32x  +  0.7424. 

.  Putting  this  equal  to  zero  and  solving  by  completing  the  square,  we 

find  

X  =  -  0.66  +  0.55V  -  1, 

in  which,  of  course,  the  coefficients  are  not  correct  to  more  than  two 
places. 

The  equation 

x^  -  lOx  -  10  =  0  (9) 

illustrates  a  case  in  which  the  cubic  has  three  real  roots.  The  straight 
line  y  =  IQx  +  10  cuts  the  cubic  parabola  (See  Fig.  95)  at  x  =  —  1.2, 
X  =  —  2.4,  and  x  =  3.6.  These,  then,  are  the  approximate  roots. 
The  product 

(x  +  1.2)  (x  +  2.4)  (x  -  3.6)  =  x'  -  lO.OSx  -  10.37 

should  give  the  original  equation  (9).  This  result  checks  the  work 
to  about  two  decimal  places. 

The  x-scale  of  Fig.  95  extends  only  from  —  6  to  +  5.  The  same 
diagram  may,  however,  be  used  for  any  range  of  values  by  suitably 
changing  the  unit  of  measure  on  the  two  scales;  thus,  the  divisions  of 
the  x-scale  may  be  marked  with  numbers  5-fold  the  present  numbers, 
in  which  case  the  numbers  on  the  y-aaaXe  must  be  marked  with  num- 
bers 125  times  as  great  as  the  present  numbers.  These  results  are 
shown  by  the  auxiliary  numbers  attached  to  the  i/-scale  in  Fig.  OS.' 

It  is  obvious  that  a  similar  process  will  apply  to  any  equation  of  the 
form 

X"  +  ox  +  6  =  0. 

Exercises 

Solve  graphically  the  following  equations  and  check  each  result 
separately : 

1.  x'  -  4x  -I-  10  =  0  4.  x'  -  15x  -  5  =  0. 

2.  x'  -  12x  -  8  =  0.  5.  x'  -  3x  +  1  =  0. 

3.  x'  -I-  X  -  3  =  0.  6.  x^  -  4x  -  2  =0. 
7.  2  sin  9  -f-  3  cos  e  =  3.5. 

»  For  other  graphical  methods  of  solution  of  equationSt  see  Runge's  Graphical 
Methods,"  Columbia  University  Press,  1912.  More  work  on  the  graphical  solution 
of  the  cubic  will  be  found  in  Schultze,  "  Advanced  Algebra,"  p.  484. 


196        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§104 

Note:  Construct  on  polar  paper  the  circles  p  =  3.5  and  p  = 
2  sin  9  +  3  cos  0. 

8.  2x  +  sin  a;  =  0.6. 

Note:  Find  the  intersec  Jon  oiy  =  sin  x  and  the  line  y=  —  2a;  +  0.6. 
If  1.15  inches  is  the  amplitude  of  3/'=  sins;,  then  1.15  inches  must  be  the 
unit  of  measure  used  for  the  construction  of  the  line  y  =  —  2a;+  0.6. 
9.  x'  +  X  +  1  +  1/x  =  0. 

10.  Show  that  a;'  +  a.T  +  b  =  0  can  have  but  one  real  root  if  a  >  0. 

104.  Method  of  Successive  Approxunations.  It  must  be  re- 
membered that  the  graphic  methods  of  solving  numerical  equa- 
tions by  finding  one  or  both  coordiaates  of  points  of  intersection  of 
graphs,  gives  results  only  approximately  correct.  The  degree  of 
accurately  depends  upon  the  scale  of  the  drawing  and  upon 
the  accuracy  with  which  the  graphs  are  constructed.  The  results 
thus  obtained  may  be  used  as  a  first  approximation  to  the  solution 
by  a  method  illustrated  below 

Suppose  that  it  is  required  to  find  to  four  decimal  places  one  root  of 
x'  —  X  —  1  =  0.     See  §103  and  Fig.  95.     The  graphic  method  gives 
X  =  1.32.     This  is  the  first  approximation.     A  second  approximation 
is  found  as  follows : 
Substituting  1.32  for  x  in 

y  =  x'  —  X  —  \  J  (1) 

gives  —  0.0200  for  y.  This  shows  that  1.32  is  not  the  exact  value  for 
y.  Substituting  1.33  for  a;  gives  0.0226  for  y.  Put  these  results  in 
tabular  form 


X 

y 

p 
Q 

1.32 
1.33 

-0.0200 
+0.0226 

Differences 

0.01 

0.0426 

This  shows  that  the  X-intercept  of  the  graph  of  the  given 
equation  is  between  the  points  P  and  Q,  Fig.  96.  Thus  a  root  of 
x'  —  a;  —  1  is  greater  than  1.32  and  less  than  1.33.  Now  reason  as 
follows:  The  actual  root  lies  between  1.32  and  1.33,  and  the  zero  value 
of  y  corresponds  to  it.  This  zero  is  approximately  200/426  of  the  way 
between  the  two  values  of  y.  Hence  if  the  curve  be  nearly  straight 
between  x  =  1.32,  and  x  =  1.33,  the  desired  value  of  x  is  approxi- 
mately 200/426  of  the  way  between  1.32  and  1.33  or  it  is  x  =  1.3247 
approximately.     This  value  is  probably  correct  to  the  fourth  decimal 


§104]       SINGLE  AND  SIMULTANEOUS  EQUATIONS        197 

place.     The  next  step  will  show  that  this  result  is  correct  to  four 
decimal  places. 

To  find  a  third  approximation  we  build  another  table  of  values: 


1.3247 
1.3248 


Differences       0.0001 


y 

-0.0000766 
+0.0003499 


0.0004265 


Fig.  96. — Method  of  approximation  to  a  root  of  an  equation. 

Reasoning  as  before,  we  get  x  =  1.324718  which  is  very  likely  true 
to  the  last  decimal  place. 

The  above  method  is  applicable  to  an  equation  like  exercise  8 
above.  In  fact  it  is  the  only  numerical  method  that  is  applicable  tn 
such  cases. 

Exercises 


Find  correct  to  four  decimal  places  the  roots  of: 

1.  x'  -ix  +  10  =  0. 

2.  X'  -  12x  -8=0.     See  Exercises  1  and  2,  §103. 


CHAPTER  VII 

PERMUTATIONS  AND  COMBINATIONS; 

THE  BINOMIAL  THEOREM  ] 

105.  Ftmdamental  Principle.  If  one  thing  can  be  done  in  n 
different  ways  and  another  thing  can  be  done  in  r  different  ways, 
then  both  things  can  be  done  together,  or  in  succession,  in  n  Xr 
different  ways.  This  simple  theorem  is  fundamental  to  the  work 
of  this  chapter.  To  illustrate,  if  there  be  3  ways  of  going  from 
Madison  to  Chicago  and  7  ways  of  going  from  Chicago  to  New 
York,  then  there  are  21  ways  of  going  from  Madison  to  New  York. 

To  prove  the  general  theorem,  note  that  if  there  be  only  one 
way  of  doing  the  first  thing,  that  way  could  be  associated  with 
each  of  the  r  ways  of  doing  the  second  thing,  making  r  ways 
of  doing  both.  That  is,  for  each  way  of  doing  the  first,  there  are 
r  ways  of  doing  both  things;  hence,  for  n  ways  of  doing  the  first 
there  are  n  X  r  ways  of  doing  both. 

Illustrations:  A  penny  may  fall  in  2  ways;  a  common  die  may 
fall  in  6  ways;  the  two  may  fall  together  in  12  ways. 

In  a  society,  any  one  of  9  seniors  is  eligible  for  president  and  any  one 
of  14  juniors  is  eligible  for  vice-president.  The  number  of  tickets 
possible  is,  therefore,  9  X  14  or  126. 

I  can  purchase  a  present  at  any  one  of  4  shops.  I  can  give  it  away 
to  any  one  of  7  people.  I  can,  therefore,  purchase  and  give  it  away  in 
any  one  of  28  different  ways. 

A  product  of  two  factors  is  to  be  made  by  selecting  the  first  factor 
from  the  numbers  a,  b,  c,  and  then  selecting  the  second  factor  from  the 
numbers  x,  y,  z,  u,  v.    The  number  of  possible  products  is,  therefore,  15. 

If  a  first  thing  can  be  done  in  n  different  ways,  a  second  in  r 
different  ways,  and  a  third  in  s  different  ways,  the  three  things 
can^be  done  in  n  X  r  X  s  different  ways.  This  follows  at  once 
from  the  fundamental  principle,  since  we  may  regard  the  first 

198 


§106]  PERMUTATIONS  AND  COMBINATIONS  199 

two  things  as  constituting  a  single  thing  that  can  be  done  in  nr 
ways,  and  then  associate  it  with  the  third,  making  nr  X  s  ways 
of  doing  the  two  things,  consisting  of  the  first  two  and  the  third. 

In  the  same  way,  if  one  thing  can  be  done  in  n  different  ways,  a 
second  in  r  different  ways,  a  third  in  s,  a  fourth  in  t,  etc.,  then  all 
can  be  done  together  inn  X  r  X  s  X  t  "different  ways. 

Thus,  n  different  presents  can  be  given  to  x  men  and  a  women 
in  (x  +  a)"  different  ways.  For  the  first  of  the  n  presents  can 
be  given  away  in  (x  +  a)  diiferent  ways,  the  second  can  be  given 
away  in  (x  +  a)  different  ways,  and  the  third  in  (a;  +  a)  different 
ways  and  so  on.  Hence,  the  number  of  possible  ways  of  giving 
away  the  n  presents  to  {x  +  a)  men  and  women  is 

(a;  +  a){x  +  a)(x  +  a)  to  n  factors,  or  {x  +  a)". 

Exercises 

1.  A  building  has  6  exits.  In  how  many  ways  can  a  person  leave 
the  building  and  enter  by  a  different  door? 

2.  A  car  has  five  seats.  In  how  many  different  ways  may  three 
people  be  seated,  each  occupying  a  different  seat? 

3.  In  how  many  different  ways  may  3  presents  be  given  away  to 
10  people? 

106.  Definitions.  Every  distinct  order  in  which  objects 
may  be  placed  in  a  line  or  row  is  called  a  permutation,  or  an 
arrangement.  Every  distinct  selection  of  objects  that  can  be 
made,  irrespective  of  the  order  in  which  they  are  placed,  is  called 
a  combination,  or  group. 

Thus,  if  we  take  the  letters  a,  b,  e,  two  at  a  time,  there  are  six 
arrangements,  namely,  ab,  ac,  ba,  be,  ca,  cb,  but  there  are  only 
three  groups,  namely,  ab,  ac,  be. 

If  we  take  the  three  letters  all  ,at  a  time,  there  are  six  arrange- 
ments possible,  namely,  abc,  acb,  boa,  baa,  cab,  eba,  but  there  is 
only  one  group,  namely,  abc. 

Permutations  and  combinations  are  both  results  of  mode  of 
selection.  Permutations  are  selections  made  with  the  understand- 
ing that  two  selections  are  considered  as  different  even  though 
they  differ  in  arrangement  only;  combinations  are  selections  made 
with  the  understanding  that  two  selections  are  not  considered  as 
different,  if  they  differ  in  arrangement  only. 


200        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§107 

In  the  following  work,  products  of  the  natural  numbers  like 

1X2X3;     1X2X3X4X5;    etc. 

are  of  frequent  occurrence.  These  products  are  abbreviated  by 
the  sjonbols  3\,  5 Land  read  "factorial  three,"  "factorial  five" 
respectively. 

107.  Formula  for  the  Number  of  Permutations  of  n  Different 
Things  Taken  All  at  a  Time.  We  are  required  to  find  how  many 
possible  ways  there  are  of  arranging  n  different  things  in  a  line. 
Lay  out  a  row  of  n  blank  spaces,  so  that  each  may  receive  one  of 
these  objects,  thus: 

I     1     I     I     2     I     I     3     I     I     4     I     I     5    I     .   .   .  MlJ 

In  the  fijst  space  we  may  place  any  one  of  the  n  objects;  therefore, 
that  space  may  be  occupied  in  n  different  ways.  The  second 
space,  after  one  object  has  been  placed  in  the  first  space,  may  be 
occupied  in  (n  —  1)  different  ways;  hence,  by  the  fundamental 
principle,  the  two  spaces  may  be  occupied  in  n(n  —  1)  different 
ways.  In  like  manner,  the  third  space  may  be  occupied  in  (n  —  2) 
different  ways,  and,  by  the  same  principle,  the  first  three  spaces 
may  be  occupied  in  n(n  —  l)(n  —  2)  different  ways,  and  so  on. 
The  next  to  the  last  space  can  be  occupied  in  but  two  different 
ways,  since  there  are  but  two  objects  left,  and  the  last  space 
can  be  occupied  in  but  one  way  by  placing  therein  the  last  re- 
maining object.  Hence,  the  total  number  of  different  ways  of 
occupying  the  n  spaces  in  the  row  with  the  n  objects  is  the  product 

n(n  -  l)(n  -2)       .    .  3-2-1, 
or 

n!. 

If  we  use  the  symbol  Pn  to  stand  for  the  number  of  permutations 
of  n  things  taken  all  at  a  time,  then  we  write 

P„  =  n!  (1) 

108.  Formula  for  the  Nxmiber  of  Permutations  of  n  Things 
Taken  r  at  a  Time.  We  are  required  to  find  how  many  possible 
ways  there  are  of  arranging  a  row  consisting  of  r  different  things, 


§108]  PERMUTATIONS  AND  COMBINATIONS  201 

when  we  may  8ele(}t  the  r  things  from  a  larger  group  of  n  different 
things. 

For  convenience  in  reasoning,  lay  out  a  row  of  r  blank  spaces, 
so  that  each  of  the  spaces  may  receive  one  of  the  objects,  thus: 

\     1     I     I     2         \     3     j  .    .    .  i  r-1  I     I     r     \ 

In  the  first  space  of  the  row,  we  may  place  any  one  of  the  n  objects; 
therefore,  that  space  may  be  occupied  in  n  different  ways.  The 
second  space,  after  one  object  has  been  placed  in  the  first  space, 
may  be  occupied  in  (w  —  1)  different  ways;  hence,  by  the  fun- 
damental principle,  the  two  spaces  may  be  occupied  in  n{n  —  1) 
different  ways.  In  like  manner,  the  third  space  may  be  occupied 
in  (n  —  2)  different  ways;  hence,  the  first  three  may  be  occupied 
in  n{n  —  !)(«■  —  2)  different  ways,  and  so  on.  The  last,  or  rth, 
space  can  be  occupied  in  as  many  different  ways  as  there  are 
objects  left.  When  an  object  is  about  to  be  selected  for  the  rth 
space,  there  have  been  used  (r  —  1)  objects  (one  for  each  of  the 
(r  —  1)  spaces  already  occupied).  Since  there  were  n  objects  to 
begin  with,  the  number  of  objects  left  is  n  —  (r  —  1),  orn  —  r  +  1, 
which  is  the  number  of  different  ways  in  which  the  last  space 
in  the  row  may  be  occupied.     Hence,  the  formula: 

P„,.  =  n(n  -  i)(n  -  2)  (n  -  r  +  i),  (1) 

in  which  P„,r  stands  for  the  number  of  permutations  of  n  things 
taken  r  at  a  time. 

This   formula,   by   multiplication   and   division   by    (n  —  r) ! 
becomes : 

_  n(n  -  1)  .   .   .  (w  -  r  +  l)(n  -  r){n  -  r  -  1)  .   .   .  3-2-1 
""■  ~  {n-r){n-r-l).  3-21 

n' 

or  P.,.  =  , v.-  (2) 

'        (n  —  r) !  ^  ' 

This  formula  is  more  compact  than  the  form  (I)  above,  but  the 
fraction  is  not  in  its  lowest  terms. 

Formula  (1)  is  easily  remembered  by  the  fact  that  there  are 
just  r  factors,  beginning  with  n  and  decreasing  by  one.  Thus  we 
have 

Pio,7  =  10X9X8X7X6X5X4. 


202        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§109 

Exercises  , 

1.  How  many  permutations  can  be  made  of  six  things  taken  all  at  a 
time? 

2.  How  many  different  numbers  can  be  made  with  the  five  digits 
1,  2,  3,  4,  5,  using  each  digit  once  and  only  once  to  form  each  number? 

3.  The  number  of  permutations  of  four  things  taken  all  at  a  time 
bears  what  ratio  to  the  number  of  permutations  of  seven  things  taken 
all  at  a  time? 

4.  How  many  arrangements  can  be  made  of  eight  things  taken  three 
at  a  time? 

5.  How  many  arrangements  can  be  made  of  eight  things  taken  five 
at  a  time? 

6.  How  many  four-figure  numbers  can  be  formed  with  the  nine 
digits  1,  2,  9  without  repeating  any  digit  in  any  number? 

7.  How  many  different  signals  can  be  made  with  seven  different 
flags,  by  hoisting  them  one  above  another  five  at  a  time? 

8.  How  many  different  signals  can  be  made  with  seven  different 
flags,  by  hoisting  them  one  above  another  any  number  at  a  time? 

9.  How  many  different  arrangements  can  be  made  of  nine  ball 
players,  supposing  only  two  of  them  can  catch  and  one  pitch? 

10.  How  many  different  ways  may  the  letters  of  the  word  algebra 
be  written,  using  all  of  the  letters? 

109.  Formula  for  the  number  of  combinations,  or  groups, 
of  n  different  things  taken  r  at  a  time. 

It  is  obvious  that  the  number  of  combinations,  or  groups,  con- 
sisting of  r  objects  each  that  can  be  selected  from  n  objects,  is 
less  than  the  number  of  permutations  of  the  same  objects  taken 
r  at  a  time,  for  each  combination  or  group  when  selected  can  be 
arranged  in  a  large  number  of  ways.  In  fact,  since  there  are  r 
objects  in  the  group,  each  group  can  be  arranged  in  exactly  r\ 
different  ways.  Hence,  for  each  group  of  r  objects,  selected  from 
n  objects,  there  exists  r!  permutations  of  r  objects  each.  There- 
fore, the  number  of  permutations  of  n  things  taken  r  at  a  time,  is 
r!  times  the  number  of  combinations  of  n  objects  taken  r  at  a 
time.     Calling  the  unknown  number  of  combinations  x,  we  have 

xXrl  =  P„„  =,     ^"  ,,, 
{n  —  r)\ 

or  solving  for  x 

^         n\ 

r!(n  —  r)! 


§109]  PERMUTATIONS  AND  COMBINATIONS  203 

This  is  the  number  of  combinations  of  n  objects  taken  r  at  a  time, 
and  may  be  symbdiized 

C 5J (I) 

This  fraction  will  always  reduce  to  a  whole  number.    It  may  be 
written  in  the  useful  form 

P      _  n{n  -  l){n  -  2)   .    .    .    (n  -  r  +  1)  ,„. 

^""  ~  1X2X3.  r  '  ^''' 

It  is  easily  remembered  in  this  form,  for  it  has  r  factors  in  both 
the  numerator  and  the  denominator.  Thus  for  the  number  of 
combinations  of  ten  things  taken  four  at  a  time  we  have  four 
factors  in  the  numerator  and  denominator,  or 

„       ^  10  X  9  X  8  X  7 
^">''       1X2X3X4   ■ 

Exercises 

1.  Howmany  different  products  of  three  each  can  be  made  with  the 
five  numbers  a,  6,  c,  d,  e,  provided  each  combination  of  three  factors 
gives  a  different  product. 

2.  How  many  products  can  be  made  from  nine  different  numbers, 
by  taking  six  numbers  to  form  each  product? 

3.  How  many  products  can  be  made  from  nine  different  numbers, 
by  taking  four  numbers  to  form  each  product? 

4.  How  many  different  hands  of  thirteen  cards  each  can  be  held  at  a 
game  of  whist? 

6.  A  building  has  5  entrances.  In  how  many  ways  can  a.  person 
enter  the  building  and  leave  by  a  different  door? 

6.  In  how  many  ways  can  a  child  be  named,  supposing  that  there 
are  400  different  Christian  names,  without  giving  it  more  than  three 
names? 

7.  In  how  many  ways  can  a  committee  of  three  be  appointed  from 
six  Italians,  four  Frenchmen,  and  seven  Americans  provided  each 
nationality  is  represented? 

8.  There  are  five  straight  lines  in  a  plane,  no  two  of  which  are  par- 
allel; how  many  intersections  are  there? 

9.  There  are  five  points  in  a  plane,  no  three  of  which  are  coUinear; 
how  many  lines  result  from  joining  each  point  to  every  other  point? 

10.  In  a  plane  there  are  n  straight  lines,  no  two  of  which  are  parallel ; 
how  many  intersections  are  there? 


204        ELEMENTARY  MATHEMATICAL  ANALYSIS      [|110 

11.  In  a  plane  there  are  n  points,  no  three  of  which  are  collinear; 
how  many  straight  lines  do  they  determine? 

12.  In  a  plane  there  are  n. points,  no  three  of  which  are  collinear, 
except  r,  which  are  all  in  the  same  straight  line;  find  the  number  of 
straight  lines  which  result  from  joining  them. 

13.  In  how  many  ways  can  seven  people  sit  at  a  round  table? 

14.  In  how  many  ways  can  seven  beads  of  different  colors  be  strung 
so  as  to  form  a  bracelet? 

15.  How  many  different  sums  of  money  can  be  formed  from  a  dime, 
a  quarter,  a  half  dollar,  a  dollar,  a  quarter  eagle,  a  half  eagle,  and  an 
eagle? 

110.*  The  Arithmetical  Triangle.  In  deriving  by  actual  mul- 
tiplication, as  below,  any  power  of  a  binomial  x  +  a  from  the 
preceding  power,  it  is  easy  to  see  that  any  coeflSicient  in  the  new 
power  is  the  sum  of  the  coefficient  of  the  corresponding  term  in  the 
multiplicand  and  the  coefficient  preceding  it  in  the  multiplicand. 
Thus 

x'  +  3ax^  +  So^a;  +  a' 

X  +  a 


X*  +  3ax^  +  3aV  +    a^x 

ax'  +  3aV  +  3a'x  +  a* 
x'^  +  Aax'  +  &aV  +  Aa'x  +  a\ 

or,  retaining  coefficients  only,  we  have 

1+3+3+1 

1^  1 

1+3+3+1 

1+3+3+1 


1+4+6+4+1 


from  which  the  law  of  formation  of  the  coefficients  1,  4,  6,  .  .  . 
is  evideAt.  Hence,  writing  down  the  coefficients  of  the  powers 
of  a;  +  o  in  order,  we  have 


§in] 


PERMUTATIONS  AND  COMBINATIONS 


205 


Powers 


CoefScients 


] 

L   2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0     ] 

1    ] 

I    1 

.  2     ] 

L    2 

1 

3     ] 

I    3 

3 

1 

4     ] 

L    4 

6 

4 

1 

5     ] 

L    5 

10 

10 

5 

1 

6 

L    6 

15 

20 

15 

6 

1 

7     ] 

I   7 

21 

35 

35 

21 

7 

1 

8 

t    8 

28 

56 

70 

56 

28 

8 

1 

9     ] 

I    9 

36 

84 

126 

126 

84 

36 

9 

1 

10 

L   10 

45 

120 

210 

252 

210 

120 

45 

10 

1 

In  this  triangle,  each  number  is  the  sum  of  the  number  above  it 
and  the  number  to  the  left  of  the  latter.  Thus  84  in  the  9th  line 
equals  56  +  28,  etc.  The  triangle  of  numbers  was  used  previous 
to  the  time  of  Isaac  Newton  for  finding  the  coefficients  of  any  de- 
sired power  of  a  binomial.  At  that  time  it  was  not  suspected 
that  the  coefficients  of  any  power  could  be  made  without  first 
obtaining  the  coeflBcients  of  the  preceding  power.  Isaac  Newton, 
while  an  undergraduate  at  Cambridge,  showed  that  the  coefficients 
of  any  power  could  be  found  without  knowing  the  coefficients  of 
the  preceding  power;  in  fact,  he  showed  that  the  coefficients  of 
any  power  n  of  a  binomial  were  functions  of  the  exponent  n. 

The  above  triangle  of  numbers  is  known  as  the  arithmetical 
triangle  or  as  Pascal's  triangle. 

111.  Binomial  Expansion.  The  demonstration  of  the  binominal 
theorem  may  be  based  upon  the  following  law  of  multiplication: 
The  product  of  any  number  of  •polynomials  is  the  aggregate  of  all 
the  possible  partial  products  which  can  be  made  by  taking  one  term 
and  only  one  from  each  of  the  polynomials.  This  statement  is 
merely  a  definition  of  what  is  meant  by  the  product  of  two  or  more 
polynomials.     (See  Chapter  XV,  §305.)     Thus, 

{x  +  a){y  +  b){z  +  c)  = 

xyz  ■+  ayz  +  bxz  +  cxy  +  abz  +  box  +  cay  +  abc 


206        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§111 

Each  of  vthe  eight  partial  products  contains  a  letter  from  each 
parenthesis,  and  never  two  from  the  same  parenthesis.  The 
number  of  terms  is  the  number  of  different  ways  in  which  a  letter 
can  be  selected  from  each  of  the  three  parentheses.  In  the  present 
case  this  is,  by  §105,  2X2X2  =  8. 

Let  it  be  required  to  write  out  the  value  of  (x  +  a)",  where  x 
and  o  stand  for  any  two  numbers  and  n  is  a  positive  integer. 
That  is,  we  must  consider  the  product  of  the  n  parentheses 

(x  +  a)(x  +  a){x  +  a)  (x  +  a), 

by  the  distributive  law  stated  above. 

First.  Take  an  x  from  each  of  the  parentheses  to  form  one  of 
the  partial  products.     This  gives  the  term  x"  of  the  product. 

Second.  Take  an  a  from  the  first  parenthesis  with  an  x  from 
each  of  the  other  (n  —  1)  parentheses.  This  gives  aa;""'  as 
another  partial  product.  But  if  we  take  a  from  the  second  paren- 
thesis and  an  x  from  each  of  the  other  (n  —  1)  parentheses,  we  get 
ax"-'-  as  another  partial  product.  Likewise  by  taking  a  from  any 
of  the  parentheses  and  an  x  from  each  of  the  other  (n  —  1)  paren- 
theses, we  shall  obtain  aa;»~'  as  a  partial  product.  Hence,  the 
final  product  contains  n  terms  like  ax"~',  or,  adding  these,  we 
obtain  nax""^  as  a  part  of  the  product. 

Third.  We  may  obtain  a  partial  product  like  a^x^~'^  by  taking 
an  a  from  any  two  of  the  parentheses,  together  with  the  x's  from 
each  of  the  other  (n  —2)  parentheses.  Hence,  there  are  as  many 
partial  products  like  o^a;»"^  as  there  are  ways  of  selecting  two  a's, 
from  n  parentheses;  that  is,  as  many  ways  as  there  are  groups,  or 
combinations,  of  n  things  taken  two  at  a  time,  or 

n{n  —  1) 

r2 

Hence,  — -  a^a;""^  is  another  part  of  the  product. 

1  '^ 

Fourth.     We  may  obtain  a  partial  product  like  a'a;"~'  by  taking 

an  a  from  any  three  of  the  parentheses  together  with  the  a;'s  from 

each  of  the  other  (»  —  3)  parentheses.     Hence,  there  are  as  many 

partial  products  like  a'a;»-'  as  there  are  ways  of  selecting  three  o's 

from  n  parentheses,  that  is,  as  many  ways  as  there  are  combina- 


§111]  PERMUTATIONS  AND  COMBINATIONS  207 

tions  of  n  things  taken  three  at  a  time,  or  V9^ 

Hence,  TY^ a^x"-'  is  another  part  of  the  product. 

In  general,  we  may  obtain  a  partial  product  like  a'x"''  (where  r 
is  an  integer  <  n)  by  taking  an  a  from  any  r  parentheses  together 
with  the  x's  from  each  of  the  other  (w  —  r)  parentheses. 
Hence,  there  are  as  many  partial  products  Uke  a'x"~'  as  there  are 
ways  of  selecting  r  a's  from  n  parentheses;  that  is,  as  many  ways 
as  there  are  combinations  of  n  .things  taken  r  at  a   time,   or 

-r-T — '■ — Ti'     Hence,   -7-7 — '- — rr    a'x"''    stands    for    any    term 
r]  {n  —  r)l  '   r\  (n  ^  r)l  •' 

in  general  in  the  product  (x  +  o)". 

Finally,  we  may  obtain  one  partial  product  like  a"  by  taking  an 

a  from  each  of  the  parentheses.     Hence,  a"  is  the  last  term  in  the 

product. 

Thus  we  have  shown  that 

/     I     \  1  11  n(n  —  I )    ,      ,   , 

(x  +  a)"  =  X"  +  nax"-!  -\ — ^ a^x""^  +   .    .    . 

1-2 

+  r!(n°-r)l^''^""^+  '    +"^"-  ^^^ 

This  is  the  binomial  formula  of  Isaac  Newton.  The  right-hand 
side  is  called  the  expansion  or  development  of  the  power  of  the 
binomial. 

It  is  obvious  that  the  expansion  of  (x  —  a)"  will  differ  from  the 
above  only  in  the  signs  of  the  alternate  terms  containing  the  odd 
powers  of  a,  which,  of  course,  will  have  the  negative  sign. 


Exercises 

1..  Expand  {u  +  Sy)^.     Here  x  =  u  and  a  =  3y.     By  the  formula 
we  get 

u^  +  bu^iSy)  +  I0u\3y)'  +  lOu^iSyy  +  5u{Zyy  +  {3y)K 

Performing  the  indicated  operations,  we  obtain 

u^  +  15u'y  +  90u'y^  +  270u^y^  +  i05uy*  +  2i3yK 

Expand  each  of  the  following  by  the  binomial  formula : 


2. 

(r«- 

■  2y. 

3. 

(3b  - 

-iy. 

4. 

(c  + 

xy. 

6. 

(2a;!! 

-x)\ 

6. 

(1- 

ay. 

7. 

(-X 

+  2ay. 

14. 

(x^ 

+  x^y. 

16. 

(o-»- 

-  lyiy. 

16. 

(\/^  -  -yaby. 

208        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§112 

8.  (i  +  xy. 

9.  (62  -  c^y. 

10.  (3o  +  iy. 

11.  (5d  -  3yy. 

12.  (3a;»_-  1)*. 

13.  (Vo  +  .!;)«. 

17.  (a  +  [X  +  2/1)'. 

18.  (a  +  6  -  ?/)'• 

19.  (a;2  +  2oa;  +  a^y. 

112.    Binomial    Expansion   .for    Fractional    and     Negative 

Exponents.    It  is  proved  in  the  Calculus  that 

/•.    ,     \        1    ,  ,    n{n  —  1)    ,  ,  n(m  —  l)(n  —  2)     ,    , 

(1  ±  a;)»  =  1  ±  na;  +  -^^-^j — -x''  ±  — ~ a;'  +  .  .  . 

is  true  for  fractional  and  for  negative  values  of  n,  provided  x  is 
less  than  1  in  absolute  value.    The  number  of  terms  in  the  expan- 
sion is  not  finite,  but  is  unlimited. 
By  the  above  formula,  we  have 

V 1  -\-  X  =  i.  +  (2)  X  -\ 21 ^   +  " 3] x'  +  .  .  . 


=  1  +  (i)  X  -  (ij  X^  +  (Vff)  X'  -  (tI^)^ 

1 
2 


If  --1 


this  becomes 

V  f  =  1  +  i  -  ^T  +  \\rs  ~  "JT^"  +  •     ■ 
Therefore,  using  five  terms  of  the  expression 

\A|=  2048  ~  1.2241  approximately. 

The  square  root,  correct  to  five  figures,  is  really  1.2247.  Thus  the 
error  in  this  case  is  less  than  one-tenth  of  1  percent  if  only  five  terms 
of  the  series  be  used.  The  degree  of  accuracy  in  each  case  is  depend- 
ent both  upon  the  value  of  n  and  upon  the  value  of  x.  Obviously,  for 
a  given  value  of  n,  the  series  converges  for  small  values  of  x  more 
rapidly  than  for  larger  values. 

As  another  example,  suppose  it  is  required  to  expand  (1  —  x)~'. 
By  the  binomial  theorem 

(1  -  x)-i  =  1  +  (- 1)(  -  x)  +  ~  ^  ^~,^  ~  ^\  -  xy 

+  -'^-'-l^^-'-'h-^y+... 

=  l+x+x'+x^+.    .    . 


§113]  PERMUTATIONS  AND  COMBINATIONS  209 

If  five  terms  of  the  series  be  used,  the  error  is  -^  f  or  a;  «=  i,  or  about 
6  percent. 

113.    Approximation  Fonnulas.    If  x  be  very  small,  the  expan- 
sion of 

(1  +  a;)«  =  1  +  ns  +  -^-^1 —  x^  +       ■    ■ 

is  approximately 

(1  +  a;)"  -  1  +  nx,  (1) 

since  x^,  x'  and  all  higher  powers  of  x  are  much  smaller  than  x. 
Thus,  using  the  symbol  ^  to  express  "approximately  equals,"  we 
have,  for  example 

(1.01)3  =  1.03. 

For,  (1  +  1/100)5  _  1  +3/100. 

The  true  value  of  (1.01)'  is  1.030301,  so  that  the  approximation  is 
very  good. 

Likewise 

(i  -  x)"  ^  I  —  nx,  (2) 

if  X  be  small. 

If  X,  y,  and  z  be  small  compared  with  unity,  the  following  ap- 
proximation formulas  hold : 

(i-+x)(i+y)^  i+x-l-y,  (3) 

f^-i+x-y,  (4) 

(i-|-x)(i-hy)(i  +  z)=T=  i4-x-f-y-hz.  (5) 

The  approximation  formulas  are  proved  as  follows : 
(1  -|-  x)  (1  +y)  =  l+x  +  y  +  xy^l+x  +  y,  for  a;?/  is  small 
compared  to  x  and  y. 

,.  I  V  =  1  +  X  —  y  +  ,  ,  =  1  +  a;  —  y,  for  the  fraction  is 
small  compared  to  x  and  y. 

1  +  x)  (1  +  y)  {1  +  z)  ^  {1  +x  +  y)  (1  +  z)  ^  1  +  X  +  y  +  z 

Exercises 
1.  Explain  the  following  approximation  formulas,  in  which  |x|  <  1 

14 


210        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§113 


Vl   -  X  "5= 

(1  +x)-i^ 

(1   +  x)-i  =F 

(1  +x^)i  === 

2.  Compute  the  approximate  numerical  value  of  the  following : 

(a)  (1.03)  i  (d)   (1.05)  i 

(6)   (1.02)  (1.03)  (e)   1.02/1.03 

(c)    (1.01)(1.02)/(1.03)(1.04). 

3.  The  formula  for  the  period  of  a  simple  pendulum  is 

T  =WT7i- 

For  the  value  of  gravity  at  New  York,  this  reduces  to 


T  = 


6.253' 


in  which  I,  the  length  of  the  pendulum,  is  measured  in  inches.     This 
pendulum  beats  seconds  when 

I  =  (6.253)=i  or  39.10  inches. 

What  is  the  period  of  the  pendulum  if  I  be  lengthened  to  39.13  inches? 


Hint: 


T  = 


6.253 


^    -  "6:253"  -  6:253^^  +  ^^^ 


VT 


(1  +  h/2l). 


6.253 

Take  I  =  39.10,  and  h  =  0.03. 
Then 

?"  =  1  +■  0.03/78.20 

=  1.00038. 

A  day  contains  86,400  seconds.  The  change  of  length  would,  there- 
fore, cause  a  loss  of  32.8  seconds  per  day,  if  the  pendulum  were 
attfiched  to  a  clock, 


§114]  PERMUTATIONS  AND  COMBINATIONS  211 

4,  On  the  ocean  how  far  can  one  see  at  an  elevation  of  h  feet  above 
its  surface? 

Call  the  radius  of  the  earth  o(  =  3960  miles),  and  the  distance  one 
can  see  d,  which  is  along  a  tangent  from  the  point  of  observation  to 

the  sphere.     Since  h  is  in  feet,  and  since  a  +  Toon!  d,  and  a  are  the 

sides  of  a  right  triangle,  we  have  (o  +  ^/5280)''  =  d'  +  a\ 
or 


"[ 


'  + sis]  ■-''■+«■■ 


Expanding  the  binomial  by  the  approximation  formula  we  have 


.[ 


'+mk]  =''  +  < 


d2  =  2a;i/5280 

=  2  X  3960^/5280 

-¥, 

or 

d  =  Vp 

where  d  is  expressed  in  miles  and  h  in  feet.     See  §68,  exercise  13. 

5.  By  what  percent  is  the  area  of  a  circle  altered  if  its  radius  of 
100  cm.  be  changed  to  101  cm.? 

6.  By  what  percent  is  the  volume  of  a  sphere,  |-7ro',  altered  if  the 
radius  be  changed  from  100  cm.  to  101  cm.? 

7.  If  the  formula  for  the  horse  power  of  a  ship  is  I.H.P.  =     „„-,i 

where  S  is  speed  in  knots  and  D  is  displacements  in  tons,  what  in- 
crease in  horse  power  is  required  in  order  to  increase  the  speed  from 
fifteen  to  sixteen  knots,  the  tonnage  remaining  constant  at  5000? 
What  increase  in  horse  power  is  required  to  maintain  the  same  speed 
if  the  load  or  tonnage  be  increased  from  5000  to  5500? 

114.*  Graphical  Representation  of  the  Coefficients  of  any 
Power  of  a  Binomial.  If  we  erect  ordiaates  at  equal  intervals 
on  the  X-axis  proportional  to  the  coeflBcients  of  any  power  of  a 
binomial,  we  find  that  a  curve  is  approximated,  which  becomes 
very  striking  as  the  exponent  is  taken  larger  and  larger.  In  Fig. 
97  the  ordinates  are  proportional  to  the  coefficients  of  the  999th 
power  of  {x  +  a).     The  drawing  is  due  to  Quetelet. 


212        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§114 

The  limit  of  the  broken  line  at  the  top  of  the  ordinates  in  Eig. 
97  is,  as  n  is  increased  indefinitely,  a  beU-shaped  curve,  known  as 


Fig.  97. — Graphical  representation  of  the  values  of  the  binomial 
coefficients  in  the  999th  power  of  a  binomial.  The  middle  coeflScients 
are  taken  equal  to  5,  for  convenience,  and  the  others  are  expressed 
to  that  scale  also. 

the  probability  curve.     In  treatises  on  the  Theory  of  Probability, 
it  is  shown  that  the  equation  of  the  curve  is  2/=ae~*^^ 


CHAPTER  VIII 
PROGRESSIONS 

116.  An  Arithmetical  Progression  or  an  Arithmetical  Series, 

is  any  succession  of  terms  such  that  each  term  differs  from  that 
immediately  preceding  by  a  fixed  number  called  the  common 
difference.    The  following  are  arithmetical  progressions: 

(1)  1,  2,  3,  4,  5. 

(2)  4,  6,  8,  10,  12. 

(3)  32,  27,  22,  17,  12. 

(4)  2i,  3i  5,  6i  7i. 

(5)  (u  -  v),  u,  {u  +  v). 

(6)  a,  a  +  d,  a  +  2d,  a  -\-  3d,   .    .    . 

The  first  and  last  terms  are  called  the  extremes,  and  the  other 
terms  are  called  the  means. 

Where  there  are  but  three  numbers  in  the  series,  the  middle 
number  is  called  the  arithmetical  mean  of  the  other  two.  To 
find  the  arithmetical  mean  of  the  two  numbers  a  and  5,  proceed  as 
follows: 

Let  A  stand  for  the  required  mean;  then,  by  definition 

A  —  a  =  b  —  A, 
whence 

A   -  ^  +  ^ 

Thus,  the  arithmetical  mean  6f  12  and  18  is  15,  for  12, 15, 18  is  an 
arithmetical  progression  of  common  difference  3. 

By  the  arithmetical  mean,  or  arithmetical  average,  of  several 
numbers  is  meant  the  result  of  dividing  the  sum  of  the  numbers 
by  the  number  of  the  numbers.    It  is,  therefore,  such  a  number 

213 


214        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§116 

that  if  all  numbers  of  the  set  were  equal  to  the  arithmetical  mean, 
the  sum  of  the  set  would  be  the  same. 

The  general  arithmetical  progression  of  n  terms  is  expressed  by: 
Number  of 

term:  12  3  4  .  n 

Progression:  a,  (a  +  d),  (a  +  2d),  (a  +  3d),  .    .    .   (a  +  [n  —  1]  d) 

Here  a  and  d  may  be  any  algebraic  numbers  whatsoever,  integral 
or  fractional,  rational  or  irrational,  positive  or  negative,  but  n 
must  be  a  positive  integer.  When  the  common  difference  is  nega- 
tive, the  progression  is  said  to  be  a  decreasing  progression ;  other- 
wise, it  is  an  increasing  progression. 

From  the  general  progression  written  above,  we  see  that  a  for- 
mula for  the  nth  term  of  any  arithmetical  progression  may  be 
written 

I  =  a  -H  (n  -  i)d,  (1) 

in  which  I  stands  for  the  nth  term. 

Formula  (1)  enables  us  to  obtain  the  value  of  any  one  of  the  num- 
bers, I,  a,  n,  d,  when  the  other  three  are  given.     Thus: 

(1)  Find  the  100th  term  of 

3  4-  8  -h  13  -I-  .    .    . 
Here  a  =  3,  d  =  5,  n  =  100. 

Therefore   '  Z  =  3  +  99  X  5  =  498. 

(2)  Find  the  number  of  terms  in  the  progression 

5  +  7  -I-  9  +  .    .    .  +  39. 
Here  a  =  5,  d  =  2,1  =  39. 

Therefore  39  =  5  -t-  (ra  -  1)2, 

or  n  =  18. 

(3)  Find  the  common  difference  in  a  progression  of  fifteen  terms  in 
which  the  extremes  are  f  and  425. 

Here  u,  =  ^,1  =  42^,  n  =  15, 

whence  42|  =  J  -F  (15  -  l)d, 

or  d  =  3. 

116.  The  Sum  of  n  Terms.  If  s  stands  for  the  sum  of  n  terms 
of  an  arithmetical  progression,  and  if  the  sum  of  the  terms  be 


§116]  PROGRESSIONS  215 

written  first  in  natural  order,  and  again  in  reverse  order,  we  have 

s  =  a  + (a  +  d)  +  (a  +  2d)  +  +  (a  +  [n  -  1]  d),       (1) 

s  =  1+  {I  -  d)  +  {I  -  2d)  +      .  .  +  Q  -In-  l]d).       (2) 

Adding  (1)  and  (2),  term  by  term,  noting  that  the  positive  and 
negative  common  differences  nullify  one  another,  we  obtain 

2s  =  (a  +  Z)  +  (a  +  Z)  +  (a  +  0  +  .   ■   ■  +  (a  +  l),         (3) 

or,  since  the  number  of  terms  in  the  original  i5rogression  is  n,  we 
may  write 

2s  =  n{a  +  I), 

or  s  =  n(a  +  l)/2.  (4) 

If  the  value  for  I,  from  (1)  §115,  be  substituted  in  formula 
(4)  it  becomes 

s  =  n  [2a  +  (n  -  i)d].  (5) 

In  equation  (4),  (a  +  Z)/2  is  the  average  of  the  first  and  nth 
terms.  The  formula  (4)  states,  therefore,  that  the  sum  equals  the 
number  of  the  terms  multiplied  by  the  average  of  the  first  and  last. 

An  arithmetical  progression  is  a  very  simple  particular  instance 
of  a  much  more  general  class  of  expressions  known  in  mathematics 
as  series.  A  series  is  any  sequence  of  terms  formed  accord- 
ing to  some  law,  such  as: 

(x  +  1)  +  (x  +  2y+  {x  +  sy  +.  .  . 

x  +  3x^  +  5x^+  .   .   . 

cos  X  +  cos  2x  +  cos  3x  -\-  .   .   . 

It  is  only  in  a  very  limited  number  of  cases  that  a  short  expression 
can  be  found  for  the  sum  of  n  terms  of  a  series.  An  arithmetical 
progression  is  one  of  these  cases. 

Formula  (4)  enables  us  to  find  the  value  of  any  one  of  the  numbers 
s,  n,  a,  I,  when  the  values  of  the  other  three  are  given.     Thus: 

(1)  Find  the  number  of  terms  in  an  arithmetical  progression  in 
which  the  first  term  is  4,  the  last  term  22,  and  the  sum  91. 

Here  a  =  4,  Z  =  22,  s  =  91, 

whence,  91  =  ra(4  +  22) /2, 

or  n  =  7. 


216        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§116 

The  two  formulas,  (1)   §116  and  (4)   §118,  contain  five  letters; 
'  hence,  if  any  two  of  them  stand  for  unknown  numbers,  and  the  values 
of  the  others  are  given,  the  values  of  the  two  unknown  numbers  can  be 
found  by  the  solution  of  a  system  of  two  equations.    Thus : 

(2)  Find  the  number  of  terms  in  a  progression  whose  sum  is  1095,  if 
the  first  term  is  38  and  the  difference  is  5. 

Here       ■  s  =  1095,  a  =  38,  and  d  =  5, 

whence,  I  =  38  +  {n  -  1)5,  (6) 

1096  =  n(38  +  l)/2.  (7) 

From  (6)  /  =  33  +  5n.  (8) 

From  (7)  2190  =  38ra  +  nl.  (9) 

Substituting  the  value  of  /  from  (8)  in  (9),  we  get 

2190  =  71n  +  5nK  (10) 

Solving  this  quadratic  equation,  we  find 

n  =  15,  or  -  29.2. 

The  second  result  is  inadmissible,  since  the  number  of  terms  cannot 
be  either  negative  or  fractional. 

Exercises 

Solve  each  of  the  following: 

1.  Given,  o  =  7,  d  =  4,  n  =  15;  find  2  and  s. 

2.  Given,  a  =  17,1  =  350,  d  =  9;  find  n  and  s. 

3.  Given,  a  =  3,  n  =  50,  s  =  3825;  find  I  and  d. 

4.  Given,  s  =  4784,  a  =  41,  d  =  2;  find  Zand  n. 

5.  Given,  s  =  1008,  d  =  4,  Z  =  88;  find  a  and  n. 

6.  Find  the  sum  of  the  first  n  even  numbers. 

7.  Find  the  sum  of  the  first  n  odd  nvmibers. 

8.  Insert  nine  arithmetical  means  between  —7/8  and  +  7/8. 

9.  Sum  (o  +  6)2  +  [a"  +  ¥)  +  (,a  -byton  terms. 

10.  Find  the  sum  of  the  first  fifty  multiples  of  7. 

11.  Find  the  amount  of  $1.00  at  simple  interest  at  5  percent  for 
1920  years. 

12.  How  long  must  $1.00  accumulate  at  3|  percent  simple  interest 
until  the  total  amounts  to  $100? 

13.  How  many  terms  of  the  progression  9  +  13  +  17  +   .    . 
must  be  taken  in  order  that  the  sum  may  equal  624?     How  many 
terms  must  be  taken  in  order  that  the  sum  may  exceed  750? 


§117]  PROGRESSIONS  217 

14.  Show  that  the  only  right  triangles  whose  sides  are  in  arithmetical 
progression  are  those  whose  sides  are  proportional  to  3,  4,  and  6. 

117.  A  geometrical  progression  or  a  geometrical  series  is  any 

succession  of  terms  such  that  each  term  is  the  product  of  the 
preceding  term  by  a  fixed  factor  called  the  ratio.  The  following 
are  examples: 

(1)  3,  6,  12,  24,  48.  (3)  1/2,  1/4,  1/8,  1/16,  1/32. 

(2)  100,  -50,  25,  -12i  (4)  a,  ar,  ar\  ar\  ar*  .       . 

The  geometrical  mean  G  of  two  numbers,  a  and  6,  is  a  number- 
such  that  a,  G,  6  is  a  geometrical  progression.     By  definition 

G/a  =  b/G, 
whence, 

G^  =  ab, 
or 

G  =  Vab. 

Thus,  4  is  the  geometrical  mean  of  2  and  8.  The  arithmetical 
mean  of  2  and  8  is  5.  The  geometrical  mean  of  n  positive  num- 
bers is  the  value  of  the  nth  root  of  their  product.  Thus  the  geo- 
metrical mean  of  8,  9,  and  24  is  -?/  8  X  9  X  24  =  12. 

118.  The  nth  Term  and  the  Sum  of  n  Terms.  If  a  represents 
the  first  term  and  r  the  ratio  of  any  geometrical  progression,  the 
progression  may  be  written: 

Number  of  term:      123      4      ..       n— 1      n. 
Progression:  o,  ar,  ar^,  ar',       .    .  ar"'^,  ar"~^. 

Therefore,  representing  the  nth  term  by  I,  we  obtain  the  simple 
formula 

1  =  ar»-i.  (1) 

Representing  by  s  the  sum  of  n  terms  of  any  geometrical  pro- 
gression, we  have 

s  =  a  -\-  ar  +  ar^  +   .    .    .   +  ar" ~^  +  ar" ~ ^, 
or, 

s  =  ail+r  +  r^+   .    .       +  r"-^  +  r"-^). 

But,  by  a  fundamental  theorem  in  factoring,  ^  the  expression  in  the 

1  See  Appendix,  Chapter  XV. 


218        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§118 


(2) 


parenthesis  is  the  quotient  of  1  —  r»  by  1  —  r.    Hence, 

a(i  —  r») 

Another  form  is  obtained  by  introducing  I  by  the  substitution 

or»-'  =  I, 

a  —  rl 

which  gives  s  =  — — —  (3) 

Formula  (1),  or  (2),  enables  one  to  find  any  one  of  the  four 
numbers  involved  in  the  equations  when  three  are  given.  The 
two  formulas  (1)  and  (2)  considered  as  simultaneous  equations 
enable  one  to  find  any  two  of  the  five  numbers  a,  r,  n,  I,  s,  when  the 
other  three  are  given.  But  if  r  be  one  of  the  unknown  numbers, 
the  equations  of  the  system  may  be  of  a  high  degree  and  beyond 
the  range  of  Chapter  VI  unless  solved  by  graphical  means.  If 
n  be  an  unknown  number,  an  equation  of  a  new  type  is  introduced, 
namely,  one  with  the  unknown  number  appearing  as  an  exponent. 
Equations  of  this  type,  known  as  exponential  equations,  will  be 
treated  in  the  chapter  on  logarithms.  The  following  examples 
illustrate  cases  in  which  the  resulting  single  and  simultaneous 
equations  are  readily  solved. 

(IJ  Insert  three  geometrical  means  between  31  and  496. 
Here 

a  =  3l,l  =  496,  and  n  =  6. 
Hence, 

496  =  31  X  r' 

r*  =  16, 
or 

r  =  ±  2. 

Consequently  the  required  means  are  either  62, 124,  and  248,  or  —  62, 
+  124,  and  -  248. 

(2)  Find  the  sum  of  a  geometrical  progression  of  five  terms,  the 
extremes  being  8  and  10,368. 
Here 

a  =  8,1  =  10,368,  and  n  =  5. 
Hence, 

10,368  =  8r*  (1) 


§118]  PROGRESSIONS  21,9 

aad 

8  =  (10,368r  -  8)/{r  -  1).  (2) 

From  the  first, 

r  =  6 
whence,  from  the  second, 

s  =  12,440. 

(3)  Find  the  extremes  of  a  geometrical  progression  whose  sum  is  635, 
if  the  ratio  be  2  and  the  number  of  terms  be  7. 

Here 

s  =  635,  r-  =  2,  and  n  =  7. 
Hence, 

I  =  a2«,  (3) 

635  =  (2/  -  a).  (4) 

Substituting  I  from  (3)  in  (4),  we  get 

635  =  128  o  -  a. 
Hence, 

a  =  5,  and  I  =  320. 

(4)  The  fourth  term  of  a  geometrical  progression  is  4,  and  the 
sixth  term  is  1.     What  is  the  tenth  term? 

Here 

ar^  =  4,  (5) 

and 

ar^  =  1.  (6) 

Dividing  (6)  by  (5)  we  obtain 


r 


2  _   1 


i,  or  r  =  +  i 
Therefore,  from  (5), 

a  =  4^/r^  =  +32. 
Then  the  tenth  term  is 

±  32(+  \y  =  tV. 


Exercises 

1.  Find  the  sum  of  seven  terms  of  4  +  8  +  16  +  .    .    . 

2.  Find  the  sum  of  -  4  +  8  -  16  +  .    .    .  to  six  terms. 

3.  Find  the  tenth  term  and  the  sum  of  ten  terms  of  4  —  2  +  1  ■ 

4.  Find  r  and  s;  given  a  =  2,1  =  31,250,  Ji  =  7. 


220        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§119 

6.  Insert  two  geometrical  means  between  47  and  1269. 

6.  Insert  three  geometrical  means  between  2  and  3. 

7.  Insert  seven  geometrical  means  between  o'  and  6*. 

8.  Show  that  the  quotient  (o"  —  6»)/(o  —  6)  is  a  geometrical 
progression. 

9.  Sum  x"""^  +   x"~'  y  +  a;""'  y'  +      .    .  to  n  terms. 

10.  Sum  a;"~i  —  a^~'  y  +  s""'  y'  —  ■    ■    .  to  w  terms. 

11.  Sum  a  +  ar~^  +  ar~'  +  .    .    .  to  n  terms. 

12.  If  a,  b,  c,  d,  ' .  .  .  are  in  geometrical  progression,  show  that 
a'^  +  6',  6*  +  c^,  c^  +  (i^  .        .  are  also  in  geometrical  progression. 

13.  If  any  numbers  are  in  geometrical  progression,  show  that  their 
differences  are  also  in  geometrical  progression. 

14.  A  man  agreed  to  pay  for  the  shoeing  of  his  horse  as  follows: 
1  cent  for  the  first  naU,  2  cents  for  the  second  nail,  4  cents  for  the  third 
nail,  and  so  on  until  the  eight  naUs  in  each  shoe  were  paid  for.  What 
did  the  last  nail  cost?.    How  much  did  he  agree  to  pay  in  all? 

119.  Compound  Interest.  Just  as  the  amount  of  principle  and 
interest  of  a  sum  of  money  at  simple  interest  for  n  years  is  ex- 
pressed by  the  (n  +  l)st  term  of  an  arithmetical  progression,  so, 
in  a  similiar  way,  the  amount  of  any  sum  at  compound  interest  for 
n  years  is  represented  by  the  (n  +  l)st  term  of  a  geometrical  pro- 
gression. Thus,  the  amount  of  $1.00  at  compound  interest  at 
4  percent  for  twenty  years  is  given  by  the  expression 

1(1.04)2". 
The  amount  of  p  dollars  for  n  years  at  r  percent  is 

K'  +  i5-o)"- 

The  present  value  of  $1.00,  due  twenty  years  hence,  estimating 
compound  interest  at  4  percent,  is 

1/(1.04)2". 

The  value  of  $1.00,  paid  annually  at  the  beginning  of  each  year 
into  a  fund  accumulating  at  4  percent  compound  interest,  is,  at 
the  end  of  twenty  years 

(1.04)1  +  (104)''  +  .    .    .   (1.04)2", 

which  is  the  sum  of  the  terms  of  a  geometrical  progression  of 
twenty  terms. 


§120]  PROGRESSIONS  221 

Problems  of  this  character  in  compound  interest,  in  compound 
discount,  and  in  the  more  complicated  problems  that  proceed 
therefrom,  are  basal  to  the  theory  of  annuities,  life  insurance,  and 
depreciation  of  machinery  and  structures.  The  computation  of 
the  high  powers  involved  necessitates  the  postponement  of  such 
problems  until  the  subject  of  logarithms  has  been  explained. 

120.  Infinite  Geometrical  Progressions.  If  the  ratio  of  a 
geometrical  progression  be  a  proper  fraction,  the  progression  is 
said  to  be  a  decreasing  progression.    Thus, 

1      1     i     i        1        cnA    ill  1 

■I)  2)  i!  8)  iw>  ana  3,  s,  jt,  ^t 
are  decreasing  progressions.     If  we  increase  the  number  of  terms 
in  the  first  of  these  progressions  the  sums  will  always  be  less  than 
2;  but  the  difference  (2  —  s)  will  become  and  remain  less  than  any 
pre-assigned  number. 

Definition:  A  constant,  a,  is  called  the  limit  of  a  variable, 
t,  if,  as  t  runs  through  a  sequence  of  numbers,  the  difference 
(a  —  t)  becomes,  and  remains,  numerically  smaller  than  any 
pre-assigned  number. 

By  definition,  2  is,  therefore,  the  limit  of  the  first  of  the  above 
progressions.     The  sum  of  n  terms  of  this  particular  progression 
should  be  written  down  by  the  student  for  a  number  of  successive 
values  for  n,  thus: 
Number  of  terms: 

1,      2,         3,         4,  5,         ...  10, 

Sum:  1,  1  +  i  1  +  f ,  1  +  I,  1  +  li  .    .       1+Ui, 

The  nth  term  differs  from  2  by  only  l/2»-i. 

It  is  easy  to  show  that  the  sum  of  every  decreasing  geometrical 
progression  approaches  a  fixed  limit  as  the  number  of  terms 
becomes  infinite.     Write  the  formula'^ 


in  the  form 

If  we  suppose  that  r  is  a  proper  fraction  and  that  n  increases  with- 


s 

= 

a  — 

or» 

1  - 

-  r 

0 

ar' 

222        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§121 

out  limit,  then  r»  can  be  made  less  than  any  assigned  number;  for, 
the  value  of  any  power  of  a  proper  fraction  decreases  as  the  ex- 
ponent of  the  power  increases.  As  the  other  parts  of  the  second 
fraction  in  (1)  do  not  change  in  value  as  n  changes,  the  fraction 
as  a  whole  can  be  made  smaller  than  any  number  that  can  be 
assigned.     Hence,  we  write 


limit 
n—  00 


b]'Th  <^' 


The  left-hand  side  is  read:  "The  limit  of  s  as  n  becomes 
infinite."     The  symbol  =  means:  "approaches"  or  "becomes." 

Exercises 
As  n  =  00 ,  find  the  limit  of  each  of  the  following : 

1.  *  -  i  + 1  -  tV  +    •  • 

Here 

a  =  -^jT  =  —  -3, 

1 

whence,  limit  s  = — j-r  =  f . 

1  ~  (  "2) 

2.  0.3333       .    . 

Here  a  =  -,%,  r  =  ,Vi 

3 
whence,  limit  s  =  —  =  \ 

3.  9-6+4-  '  ^.\-\+-h---- 

4.  0.272727   ...  7.  4  4-  0.8  +  0.16  +  .    .    . 

5.  0.279279279   .    .    . 

8.  Express  the  number  8  as  the  sum  of  an  infinite  geometrical 
progression  whose  second  term  is  2. 

121.*  Graphical  Representation.  Note  that  all  the  essentials 
of  a  geometrical  progression  may  be  studied  if  we  assume  the 
first  term  to  be  unity,  for  the  number  a  occurs  only  as  a  single 
constant  multiplier  in  each  term,  and  also  occurs  in  the  same 
manner  in  the  formulas  for  I  and  s. 

To  represent  the  geometrical  series  1  +  r  +  r''  .  .  +  r"-' 
graphically,  lay  off  OM  =  1  on  OY,  OSi  =  1  on  OX,  SiPi  = 
r  on  the  unit  line,  and  draw  MP^.  Draw  the  arc  P11S2  and  erect 
P2S2.  Draw  the  arc  P2'S2  and  erect  PzSs.  Continue  this  con- 
struction until  the  perpendicular  P„iS„  is  erected.  The  series  of 
trapezoids  OMPiSi,  S1P1P2S2,   SJ'^iPiSi,  ..  .       ,  S,_iP„_iP„S„ 


§121] 


PROGRESSIONS 


223 


are  similar  and,  since  PiSi  =  r  X  OM,  it  follows  that  P2S2  = 
rPiSiyPiSa  =  rP^Si,  .  ,  P„S„  =  rF„_iS„_i.    Hence  we  have: 

OM  =  OSi   =1 

PkSi  =  S1S2  =  r  .'.  0<Si2  =  1  +  r  =  sum  of  2  terms 


P2S2  =  O203 


,0^3  =  1  +r  +  r2 


sum  of  3  terms 


PzSs  =  S^Si  =  r^  ;.  OSi  =  1  +  r  +  r^  +  r'  =  sum  of  4  terms 


Pn-iSn-i  =  Sn-iS„  =  r"-' .-.  OSn  =  1  +  r  +r'  + 
sum  of  n  terms. 


*m— 1  = 


O         Si  S2  S3  S«  Sii 

Fig.  98. — Graphical  construction  of  the  sum  of  a  G.  P.  r  >  1. 

Fig.  98  shows  the  series  whose  ratio  is  r-  =  1.2.     Fig.  99  shows 
the  series  whose  ratio  is  0.8. 


Y 

U 

M 

P, 

P2 

Pa 

1 

^          \ 

,.X 

r^ 

f* 

Pi 

' 

1 

^ 

n 

NJTs      —- — 

- — .___^ 

O  Si  S,  Sa         Si      So  L 

Fig.  99. — Graphical  construction  of  the  sum  of  a  G.  P.  r  <  1. 

The  line  MPi  has  the  slope  (r  -  1)  in  Fig.  98  and  the  slope 
—  (1  —  r)  in  Fig.  99.    In  each  case  the  F-intercept  is  1.     Its 

1-2/ 


equation   is,    in    both    cases,  y  =  (r 
In  both   figures,   when   y  =  P^S, 


l)x  +  1,  ora;  = 


1  -r 
r",  X  =  OSn.     Substituting 
these    values   for   x  and  y,  we  get  for  the  sum  of  n  terms, 
1  —  r" 

Fig.  98  shows  that  when  the  number  of  terms  is 


1  -r 


224        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§122 

allowed  to  increase  without  limit,  the,  sum  OSn  also  increases  with- 
out limit.  Fig.  99  shows  that  when  the  number  of  terms  is  made  to 
increase  without  limit,  the  sum  0S„  approaches  OL  as  a  limit. 
Now  the  value  of  OL  is  the  value  of  x  when  ^  =  0.    Hence  the 

limit  of  the  sum  of  the  progression,  or  OL,  is  -t-^ — 
Consult  also  §9,  problem  6,  exercise  3  and  Figs.  15,  16. 
122.*  Harmonical  Progressions.    A  series  of  terms  such  that 
their  reciprocals  form  an  arithmetical  progression  are  said  to  form 
an  harmonical  progression.     The  following  are  examples: 
C1^  1  1  1    1 

('■)    2:   3>   4!  T- 

(2)  1,  T,  T)  TT- 

(3)  l/{x-y),l/x,  l/(x  +  tj). 

(4)  i  1,  -  1,  -  i 

(5)  4,  6,  12. 

(6)  1/a,  l/(a  +  d),  1/ia  +  2d),       .    . 

Although  harmonical  progressions  are  of  such  a  simple  character, 
no  simple  expression  has  been  found  for  the  sum  of  n  terms.  Our 
knowledge  of  arithmetical  progressions  enables  us  to  find  the 
value  of  any  required  term  and  to  insert  any  required  number 
of  harmonical  means  between  two  given  extremes,  as  in  the 
examples  below. 

(1)  Write  six  terms  of  the  harmonical  progression  6,  3,  2. 

We  must  write  six  terms  of  the  arithmetical  progression,  ^,  ^,  ^. 
The  common  difference  of  the  latter  is  ^,  so  that  the  arithmetical  pro- 
gression is  §,  §,  §,  f ,  ^,  1,  and  the  harmonical  progression  is  6,  3,  2, 
1.5,  1.2,  1. 

(2)  Insert  two  harmonical  means  between  4  and  2. 

We  must  insert  two  arithmetical  means  between  ^  and  -^;  these  are 
^  and  -1%,  whence  the  required  harmonical  means  are  3  and  2.4. 

123.*  Harmonical  Mean.  The  harmonical  mean  of  two 
numbers  is  found  as  follows:  Let  the  two  numbers  be  a  and  6 
and  let  H  stand  for  the  required  mean.    Then  we  have 

1/H  -  1/a  =  1/6  -  1/H. 
That  is, 

2/H  =  1/a  +  1/6  =  (a  -I-  6)  /ab. 
Hence, 

•     H  =  2ab/(a -1- b).  (1) 


§124]  PROGRESSIONS  225 

Thus  the  harmonical  mean  of  4  and  12  is  96/(4  +  12)  =  6. 
By  the  harmonical  mean  of  several  numbers  is  meant  the  reciprocal 
of  the  arithmetical  mean  of  their  reciprocals.  Thus  the  har- 
monical mean  of  12,  8,  and  48  is  13i-t- 

124.  *    Relation  between  A,  G,  and  H.    As  previously  found, 

A=  {a+  6)/2,  G=  V^,H  =  2ah/{a  +  b). 
Hence, 

AH  =  ab, 
and,  since  ab  =  C, 

AH  =  G\ 


or 


G  =  VaH.  (1) 


Exercises 

1.  Continue  the  harmonical  progression  12,  6,  4. 

2.  Find  the  difference  (1.8  +  1.2  4-  0.8  +  .  to  8  terms) 

-  (1.8  +  1.2  +  0.6  +  .    .    .  to  8  terms). 

3.  If  the  arithmetical  mean  between  two  numbers  be  1,  show  that 
the  harmonical  mean  is  the  square  of  the  geometrical  mean. 

Questions  and  Exercises  for  Review  of  Chapters  I  to  VIII 

1.  Define  scale;  uniform  scale;  non-uniform  scale;  arithmetical 
scale;  algebraic  scale;  double  scale. 

2.  Define  constant;  variable. 

3.  Define  function;  increasing  function;  decreasing  function;  even 
function;  odd  function. 

4.  Give  illustrations  of  even  functions;  of  odd  functions. 

6.  Express  the  area,  A,  of  an  equilateral  triangle  as  a  function  of  the 
length,  X,  of  its  sides. 

6.  Express  the  volume,  V,  of  a  right  circular  cone  as  a  function  of  its 
altitude  h.     The  radius  of  the  base  is  10  inches. 

7.  A  strip  of  tin  L  feet  long  and  40  inches  wide  is  made  into  a  gutter 
with  rectangular  cross  section,  by  bending  up  an  equal  portion  of  each 
side.  Express  the  cross  section,  y,  of  the  gutter  as  a  function  of  the 
breadth,  x,  of  the  amount  of  tin  turned  up.  Show  that  the  maximum 
cross  section  is  200  square  inches. 

8.  A  strip  of  tin  24  inches  square  has  an  equal  square  cut  from  each 
corner.     The  rectangular  projections  are  then  turned  up  to  form  a  tray 

15 


226        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§124 

with  square  base  and  rectangular  sides.  If  x  is  the  side  of  the  square 
cut  out  show  that  4x(12  —  x)*  is  the  function  representing  the  volume 
of  the  tray. 

9.  In  a  triangle  whose  sides  are  6,  8,  and  10  feet  is  inscribed  a  rec- 
tangle the  base  of  which  lies  in  the  longest  side  of  the  triangle.  Ex- 
press the  area,  A,  of  the  rectangle  as  a  function  of  its  altitude,  h. 

10.  A  ladder  20  feet  long  leans  against  the  vertical  wall  of  a  house. 
Express  the  area,  A,  of  the  triangle  formed  by  the  ladder,  the  wall,  and 
the  horizontal  ground,  as  a  function  of  the  distance,  x,  of  the  foot  of 
the  ladder  from  the  wall. 

11.  Find  graphically  the  values  of  the  following:  (a)  (31.6) (7.21); 

(6)  f^;    (c)  (1.36)';  (d)  ~-y 

12.  Describe  the  method  of  representing  the  position  of  points  on  a 
plane  by  the  rectangular,  or  Cartesian,  system  of  coordinates.  Define 
axes;  origin;  abscissa;  ordinate;  quadrant.  How  are  the  quadrants 
numbered? 

13.  What  is  meant  by  the  graph,  locus,  or  curve,  of  an  equation? 

14.  What  is  meant  by  the  equation  of  a  curve,  graph,  or  locus  of  a 
point. 

15.  Which  of  the  following  points  are  on  the  curve  Zx  -\-2y  =  4: 
(a)   (2,  -1);  (6)  (3,  1);  (c)  (-4,  8);  (d)  (0,  0). 

16.  Find  the  distance  of  each  of  the  following  points  from  the  origin : 
(a)  (1,  3);  (6)  (-2,  3);  (c)  (2,  -3);  (d)  (-3,  -2). 

17.  Show  that,  for  all  values  oi  m,  y  =  mx  is  a  straight  line  passing 
through  the  origin. 

18.  Show  that  the  equation  of  any  straight  line  passing  through  the 
origin  is  of  the  form  y  =  mx. 

19.  Find  the  equation  of  a  straight  line  passing  through  the  origin 
and  the  point  (—3,  5). 

20.  Show  that,  for  all  values  of  m  and  b,y  =  mx  +  6  is  the  equation 
of  a  straight  line. 

21.  Show  that  the  equation  of  any  straight  line  is  of  the  form 
y  =  mx  +  b. 

22.  Find  the  equation  of  a  straight  line  passing  through  the  points 
(1,  3)  and  (-2,  5). 

23.  Define  slope  of  a  straight  line. 

24.  Define  K-intercept,  and  X-intercept,  of  a  straight  line. 

25.  Find  the  slope,  s-intercept,  and  2/-intercept,  for  the  following: 

(a)  3x  +2y  =  6;     (6)  x  -  2y  =  5;    (c)  2y  -  3x  =  7. 

26.  Define  X-,  and  K-intercepts  of  a  curve. 


§124]  PROGRESSIONS  227 

27.  Write  the  equations  of  a  line  if: 

(a)   F-intercept  is  3  and  slope  is  2, 
(6)  y-intereept  is  1  and  slope  is  —2, 
(c)  y-intercept  is  —2,  and  slope  is  5, 
id)  X-interoept  is  3  and  slope  is  2, 
(e)  X-intercept  is  —2  and  slope  is  3, 
(J)  X-intercept  is  J  and  slope  is  —  ^, 
(g)  X-intercept  is  2  and  i/-intercept  is  3. 

28.  What  is  meant  by  curve  of  the  parabolic  type? 

29.  What  is  meant  by  curves  of  the  hyperbolic  type? 

30.  What  is  the  parabola? 

31.  What  is  the  equilateral,  or  rectangular,  hyperbola? 

32.  What  is  the  cubical  parabola? 

33.  What  is  the  semi-cubical  parabola? 

34.  When  is  a  curve  symmetrical  with  respect  to  the  X-axis;  with 
respect  to  the  y-axis;  with  respect  to  the  line  x  =  y;  with  respect  to 
the  line  y  =  —  x;  with  respect  to  the  origin?  Give  equation  of  two 
curves  for  each  of  the  cases  considered  above. 

35.  Sketch,  y  =  x^;     y  =  \x^;    y  =  2x\ 

36.  Sketch  y''  =  x;    y^  =  Jx;    y'  =  2x. 

37.  Sketch  y  =  x';    y  =  -  x'. 

1  2  1 

38.  Sketch  y  = -;    V  =  -  ^:    y  =  2i' 

39.  Sketch  x'^  =  y';    x'  =  y'^. 

40.  Sketch  y  =  x^;    y  =  —  x'. 

41.  Define  rational  equation,  empirical  equation. 

42.  Write  the  equation  of  the  curve  y  =  x^  —  3x,  after  it  is 
translated 

(a)  two  units  to  the  right;  (6)  three  units  to  the  left;  (c)  one  unit 
up;  (d)  five  units  down;  (e)  one  unit  to  the  left  and  two  units  down. 

43.  Find  the  coordinates  of  the  vertex  of : 

(o)  y  =  x''  +  2x;  (6)  y  =  x'  -  2x  +  3; 

(c)  y  =  3x^  +  6x;         (d)  y  =  6x  -  3x^  +  2. 
2;  _]_  3 

44.  Show  that  y  =  — 3^  is  an  hyperbola.     Write  the  equation  of 

its  asymptotes. 

45.  What  is  meant  ty  shearing  notion? 

46.  Show  that  shearing  the  curve  y  =  ax'  in  the  line  y  =  mx,  is 
equivalent  to  translating  the  original  curve.  Find  the  coordinates 
of  the  vertex  of  the  translated  curve. 

47.  What  is  meant  by  the  roots  of  a  function? 


228        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§124 

48.  Find  the  roots  of: 

(a)  *2  +  2x  -  3;     (6)  x^  -  3x;     (c)  3x^  +  2x  -  6. 

49.  Write  the  equation  of  the  curve  j/^  =  x'  —  x'  when  reflected  in : 
(a)  the  X-axis;  (6)  the  F-axis;  (c)  the  line  x  =  y;  (d)  the  line 
X  =   —  y. 

50.  The  roots  of  a  function  correspond  to  what  points  on  the  graph 
of  the  function? 

52.  Write  the  equation  of  a  circle,  radius  o,  center  at  the  origin; 
center  at  the  point  (h,  k). 

53.  Show  that  x'  +  y'  +  2gx  +  2/v  +  d  =  0  is  a  circle. 

54.  Find  the  coordinates  of  the  center  and  the  length  of  the  radius 
of: 

(a)  x'  +  y'  -  2x  -  ^y  +  1  =  0;   {d)  2x'  +  2y^  +  3x  +  by  =  0; 
(6)  x2  +  ^2  +  2x  +  42/  +  1  =0;    (e)  Sx^  +  S?/^  -  6x  -  V2y  =  10; 
(c)   x2  +  2/2  +  3x  -  42/  =  0;  (/)   x^  +  2/'  +  7x  -  VZy  =  25. 

66.  Which  circles  of  exercise  54  pass  through  the  origin? 
66.  Write  the  equation  of  the  circle  if  i 

(o)  the  radius  is  5  and  the  center  is  at  (1,  2) ;     ' 
(6)  the  radius  is  6  and  the  center  is  at  ( —  J^,  2) ; 

(c)  the  radius  is  10  and  the  center  is  at  ( —  2,  —  3) ; 

(d)  it  passes  through  the  origin  and  the  center  is  at  (1,  1) ; 

(e)  it  passes  through  the  origin  and  the  center  is  at  (  —  2,  3); 
(/)   it  passes  through  (1,  2)  and  the  center  is  at  (—2,  3). 

57.  Write  the  equation  of  a  line  passing  through  the  origin  and  the 
center  of  the  circle  ' 

x2  +  2/2  -  2x  +  32/  =  5. 

58.  Write  the  equation  of  a  circle  passing  through  the  point  (2,  3) 
and  through  the  center  of  the  circle 

x2  +  2/^  -  3x  -  22/  =  0. 
69.  Show  that  if  two  straight  lines  are  mutually  perpendicular,  the 
slope  of  one  is  the  negative  reciprocal  of  the  slope  of  the  other. 

60.  Show  that  {x  -  a)^  +  y^  =  a^  and  (x  -  ZaY  +  y^  =  a'  are 
tangent  to  each  other. 

61.  Find  analytically  the  coordinates  of  the  points  of  intersection 
of  x2  +  2/2  —  4x  —  92/  =  9  and  y  —  ^  x  +  1. 

62.  Find  approximate  solutions  for  exercise  61  by  drawing  the 
curves  on  squared  paper. 

63.  Solve  graphically  the  simultaneous  equations 

x^  +  y'  -  2x  ~  4:y  =  4: 
-£2  +  2/2  +  4x  -  42/  =  0. 

64.  Define  degree';  radian. 

66.  Define  the  six  circular  functions. 


§124]  PROGRESSIONS  229 

66.  Express  the  following  as  radians: 

(o)  45°;  (6)  90°;  (c)  180°;  (d)  135°;  (e)  225°;  (f)  60°;  (g)  30°;  (h)  300°; 
(i)  270°;  U)  315°;  (ft)  120°;  (l)  160°;  (ra)  216°;  (n)  310°. 

67.  Express  the  following  radians  as  degrees: 

(a)  i^;  (fe)i^;  (c)  |^;  (rf)  |^;  (e)  |^;  (/)  3;  (?)  2. 

68.  How  many  revolutions  per  minute  are  10  radians  per  second? 

5  7r  radians  per  second?    fir  radians  per  second? 

69.  A  car  is  running  at  the  rate  of  30  miles  per  hour.  Its  36-inch 
tire  is  revolving  at  the  rate  of  how  many  radians  per  second? 

70.  A  shaft  rotates  at  the  rate  of  15,000  revolutions  per  minute. 
What  is  its  angular  velocity  in  radians  per  second? 

71.  Give  the  values  of  the  circular  functions  of: 

(a)  30°;  (6)  60°;  (c)  45°. 

72.  Give  the  algebraic  signs  of  the  functions  of  an  angle  in  the  first 
quadrant;  in  the  second  quadrant ;  in  the  third  quadrant;  in  the  fourth 
quadrant. 

73.  Give  the  functions  of  the  following  angles: 

(o)  120°;  (6)  135°;  (c)  150°;  (d)  210°;  (e)  225°;  (J)  240°;  {g)  300°; 
(h)  316°;  W  330°;  (j)  0°;  (k)  90°;  (I)  180°;  (m)  270°;  (n)  360°. 

74.  Find  the  functions  of  a  if : 

(a)  sin  a  =  f  and  cos  a  is  negative; 
(6)  sin  a  =  f  and  cos  a  is  positive; 

(c)  sin  a  =  -f  and  tan  a  is  positive; 

(d)  sin  a  =  f  and  tan  a  is  negative; 

(e)  tan  a  =  2  and  cos  a  is  negative; 
(/)  tan  a  =  —  3  and  sin  a  is  positive; 
(g)  sec  or  =  5  and  tan  a  is  negative. 

75.  Which  of  the  circular  functions  are  even  functions?  Which  are 
odd  functions? 

76.  Show  that  cos  a  =  sin  (^  —  a) . 

77.  Draw  the  graph  oi  y  =  sin  x. 

78.  Show  that  the  curve  for  y  =  cos  x  may  be  obtained  from  the 
curve  for  y  =  sinx  by  translating  it  ■ir/2  units  to  the  left. 

79.  Show  that  sin^  a  +  cos^  a  =  1. 

80.  Show  that  sec^  a  =  1  +  tan"  a. 

81.  Show  that  esc"  a  =  1  +  cot"  a. 

82.  Show  that  tan  a  = 

cos  a 

nn    on  ii     i       J  cos  a 

83.  Show  that  cot  a  =  -^ • 

sm  a 


230        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§124 

84.  Express  sin  a;  in  terms  of: 

(a)  cos  x;  (6)  tan  x;  (c)  cot  x;  (d)  sec  x;  (e)  esc  x. 
86.  Express  cos  x  in  terms  of: 

(a)  sin  x;  (6)  tan  x;  (c)  cot  x;  {d)  sec  x;  (e)  esc  x. 

86.  Express  tan  x  in  terms  of: 

(a)  sin  x;  (6)  cos  x;  (c)  cot  a:;  (d)  sec  x;  (?)  esc  a;. 

87.  The  longer  leg  of  a  plot  of  land  in  the  form  of  a  60°  right 
triangle  is  80  rods.     Find  the  area  of  the  plot  in  acres. 

88.  A  plot  of  land  in  the  form  of  a  60°  right  triangle  contains 
72  acres. 

Find  the  length  in  rods  of  each  side  of  the  triangle. 

Hint  :  Let  x  represent  the  number  of  rods  in  the  length  of  the  shorter 
leg. 

89.  The  shorter  side  of  a  rectangle  is  100  feet,  the  diagonal  is  200 
feet.     Find  the  length  of  the  longer  side. 

90.  Explain  how  points  may  be  located  in  a  plane  by  means  of  polar 
coordinates. 

91.  Define  pole;  polar  axis;  radius  vector;  vectorial  angle. 

92.  Draw  curves  for: 

(a)  p  =  1;  (6)  p  =  2;  (c)  p  =  3;  (d)  p  =  5;  (e)  e  =  0;  (/)  e  =,r/4; 
(g)  e  =  7r/3;  {h)  e  =  T-/2;  6  =  2. 
Hint:  fl  is  measured  in  radians. 

93.  What  curve  is  represented  by  p  =  a  cos  9?     Prove. 

94.  What  curve  is  represented  by  p  =  6  sin  9?     Prove. 

95.  Draw   on   a   sheet  of  polar  coordinate  paper  curve  for  the 
following : 

(a)  p  =  2  cos  e;  (6)  p  =  —  2  cos  9; 

(c)  p  =  2  sin  9;  (d)  p  =  —  2  sin  9. 

96.  Prove  that  p  =  a  cos  9  +  6  sin  9  is  a  circle. 

97.  Draw  curves  for  the  following : 

(a)  p  =  2  cos  9  +  3  sin  9;  (6)  p  =  3  cos  9  —  2  sin  9; 

(c)   p  =  —  2  cos  9  +  sin  9;  (d)  p  =  —  3  cos  9  —  3  sin  9. 

98.  Draw  the  circles  p  =  1  and  p  =  cos  9  and  from  them  plot  the 
graph  for  p  =  1  +  cos  9. 

99.  Plot  curve  for  the  following  equations : 

(a)  p  =  1  +  sin  9;  (6)  p  =  1  —  sin  9; 

(c)  p  =  2  +  cos  9;  (d)  p  =  1  -  2  cos  9. 


§124]  PROGRESSIONS  231 

100.  From  a  sheet  of  polar  coordinate  paper,  form  M3,  find  values 
for  the  following : 

(a)  sin  30°;  (6)  cos  30°;  (c)  sin  45^;  (d)  cos  45°;  (e)  tan  45°;  (/)  cos  10°; 
(g)  sin  116°;  (h)  cos  216°;  (i)  sin  127°;  (j)  tan  37°;  (fc)  sin  227°; 
{I)  cos  316°. 

101.  Show  that  when  the  curve  for  p  =  f{8)  is  rotated  about  the 
pole  through  an  angle  a,  its  equation  becomes  p  =  f{e  —  a). 

102.  State  fourteen  "Theorems  on  Loci." 

103.  Find  the  polar  equation  of  a  straight  line. 

104.  The  center  of  the  circle  p  =  10  sin  (9  —  a)  lies  on  the  line 
X  —  y  =  3.     Find  a. 

105.  The  center  of  the  circle  p  =  10  cos  (0  —  a)  lies  on  the  line 
3x  -  2y  =  1.     Find  a. 

106.  The  center  of  the  circle  p  =  5  sin  {$  +  a)  lies  on  the  line 
X  -  2?/   =  6.     Find  a. 

107.  Write  the  Cartesian  equations  for: 

(a)  p  =  2  cos  e  +  3  sin  6; 
(6)  p  =  3  cos  9  —  2  sin  $; 
(c)  p  =  2  sin  9  —  3  cos  $. 

108.  Solve  analytically  2  =  2  cos  9  —  3  sin  9  for  all  values  of  8 
between  0°  and  360°. 

109.  Solve  graphically  the  equation  given  in  exercise  108. 

110.  Sketch  a  curve  for  y  =  -  —  2x. 

"        X 

111.  Sketch  a  curve  for  y  =  ^  -j-  sin  x. 

112.  A  circle  is  inscribed  in  a  30°,  60°  right  triangle.  Find  the 
diameter  of  the  circle  if  the  shorter  leg  of  the  triangle  is  4  inches;  if 
the  longer  leg  of  the  triangle  is  6  inches;  if  the  hypotenuse  of  the 
triangle  is  10  inches.  Find  the  lengths  of  the  three  sides  of  the  tri- 
angle if  the  radius  of  the  inscribed  circle  is  6  inches. 

113.  A  circle  is  inscribed  in  a  45°  right  triangle.  Find  the  diameter 
of  the  circle  if  the  legs  of  the  triangle  are  each  4  inches  in  length. 

114.  A  circle  is  circumscribed  about  a  30°,  60°  right  triangle.  Find 
the  radius  of  the  circle  if  the  hypotenuse  of  the  triangle  is  10  inches. 

116.  Write  the  polar  equation  for 

x^  -  y^  =  a2(x2  +  2/2)2. 

116.  Define  an  ellipse;  major  axis;  minor  axis. 

117.  Give  parametric  equations  for  an  ellipse. 


232        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§124 

118.  Find  the  coordinates  of  the  center  and  the  lengths  of  the  semi- 
axes  of  the  ellipse 

X  =  3  +  2  cos  a 
y  =  2  —  svD.  a. 

119.  Show  that  every  section  of  a  right  circular  cylinder  by  a  plane 
is  an  ellipse. 

120.  Show  that  the  projection  of  a  circle  upon  a  plane  is  an  ellipse. 

121.  Define  an  equilateral,  or  rectangular,\  hyperbola.  Define  an 
hyperbola. 

122.  Give  parametric  equations  for  an  hyperbola. 

123.  Define  the  axes,  the  center,  and  the  asymptotes  of  an 
hyperbola. 

124.  Find  the  coordinates  of  the  center,  the  lengths  of  the  semi- 
axes,  and  the  equations  of  the  asymptotes  for 

x'  -  y^  +  2x  -  ^y  =  11. 

126.  Find  the  equation  for  the  curve  of  sy  =  4  when  rotated  about 
the  origin  through  an  angle  of  —45° 

126.  Define  conjugate  hyperbolas. 

127.  Write  the  equation  of  the  hyperbola  conjugate  to 

x'i   —  y''   —  X  -\-  iy   =   11. 

128.  Sketch  the  curve  with  asymptotes  for 

X  =  3  +  2  sec  a 
2/  =  1  —  3  tan  a. 

129.  Write  the  equation  of  the  curve  formed  when  the  circle 
x2  -|-  j/2  =  o'  is  sheared  in  the  Une  y  =  x.     Sketch  the  curve. 

130.  Write  the  equation  of  the  curve  formed  when  the  hyperbola 

x^        V^ 

-J  —  ^  =  1  is  sheared  in  the  line  y  =  x.     Sketch  the  curve. 

131.  State  and  prove  the  remainder  theorem. 

132.  State  and  prove  the  factor  theorem. 

133.  Without  performing  the  division,  find  the  rehiainder  of 
(s'  -  2x2  +  3  -  1)  -H  (a;  +  2). 

134.  Explain  what  is  meant  by  questionable  and  legitimate 
transformations. 

136.  Explain  a  method  of  finding  approximately  the  roots  of  a 
cubic  equation. 

136.  Find  the  equation  of  the  straight  line  passing  through  the 
points  of  intersection  oi  x^  +  y'  +  2x  +  4y  —  11  —  0  and 
X-  +  y^  -  2x  -  2y  =  0. 


§124]  PROGRESSIONS  233 

137.  What  are  the  equations  of  the  coordinate  axes? 

138.  What  is  the  locus  of  x^  =  4?  of  y'  =  4?  of  a^  =  2/«? 
of  o2a;2  =  62!/2? 

140.  Solve    {  „  ,       .",      „         . 

141.  Define  series. 

142.  Define  arithmetical  progression;  geometrical  progression;  har- 
monical  progression. 

143.  Define  arithmetical  mean;  geometrical  mean. 

144.  Derive  formulas  for  I  and  s  of  an  arithmetical  progression. 
146.  Derive  formulas  for  I  and  s  of  a  geometrical  progression. 

146.  Define  an  infinite  geometrical  progression.         • 

147.  Derive  the  formula  for  the  sum  of  an  infinite  geometrical 
progression. 

148.  Find  the  value  of  0.273273273       .    . 

149.  A  debt  of  $10,000  is  to  be  paid  in  ten  years.  An  equal  amount 
is  paid  at  the  end  of  each  year.  Find  this  amount  if  the  indebtedness 
draws  interest  at  5  percent. 

150.  An  equal  amount  of  money  is  deposited  at  the  end  of  each  year 
for  twenty  years  as  a  sinking  fund  to  replace  a  piece  of  machinery 
valued  at  $10,000.  How  much  must  be  deposited  at  the  end  of  each 
year,  if  the  deposits  draw  4  percent  compound  interest. 


CHAPTER  IX 

THE  LOGARITHMIC  AND  THE  EXPONENTIAL 
FUNCTIONS 

125.  Historical  Development.  The  almost  miraculous  power 
of  modern  calculation  is  due,  in  large  part,  to  the  invention  of 
logarithms  in  the  first  quarter  of  the  seventeenth  century  by  a 
Scotchman,  John  Napier,  Baron  of  Merchiston.  This  invention 
was  founded  on  a  very  simple  and  obvious  principle,  that  had 
been  quite  overlooked  by  mathematicians  for  many  genera- 
tions. Napier'sinventionmay  be  explained  as  follows:^  Let  there 
be  an  arithmetical  and  a  geometrical  progression  which  are  to  be 
associated  together,  as,  for  example,  the  following : 

0,  1,     2,    3,     4,      5,      6,       7,        8,        9,        10 

1,  2,     4,     8,     16,     32,     64,     128,     256,     512,     1024 

Now  the  product  of  any  two  numbers  of  the  second  line  may  be 
found  by  adding  the  two  numbers  of  the  first  progression  above 
them,  finding  this  sum  in  the  first  Une,  and  finally  taking  the  num- 
ber lying  under  it ;  this^Iatter  number  is  the  product  sought.  Thus, 
suppose  the  product  of  8  by  32  is  desired.  Over  these  numbers 
of  the  second  line  stand  the  numbers  3  and  5,  whose  sum  is  8. 
Under  8  is  found  256,  the  product  desired.  Now  since  but  a 
limited  variety  of  numbers  is  offered  in  this  table,  it  would  be 
useless  in  the  actual  practice  of  multiplication,  for  the  reason 
that  the  particular  numbers  whose  product  is  desired  would 
probably  not  be  found  in  the  second  line.  The  overcoming 
of  this  obvious  obstacle  constitutes  the  novelty  of  Napier's  inven- 
tion. Napier  proposed  to  insert  any  number  of  intermediate 
terms  in  each  progression.     Thus,  instead  of  the  portion 

0,  1,    2,    3,      4 

1,  2,    4,    8,     16 

1  Merely  the  fundamental  principles  of  the  invention,  not  historical  details,  are 
given  in  what  follows.  For  a  very  brief  course  in  logarithms,  only  §§131-144 
need  be  taken. 

234 


§126]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  236 

of  the  two  series  we  may  wi-ite 

0,  I      1,       U,    2,        2i     3,  3),      4 

1,  \/2,     2,     Vs,     4,     a/32,     8,     ^128,     16 

by  inserting  arithmetical  means  between  the  consecutive  terms 
of  the  arithmetical  series  and  by  inserting  geometrical  means 
between  the  terms  of  the  geometrical  series.  Let  these  be 
computed  to  any  desired  degree  of  accuracy,  say  to  two  decimal 
places.    Then  we  have  the  series 


A.  P. 

G.P. 

0.0 

1.00 

0.5 

1.41 

1.0 

2.00 

1.5 

2.83 

2.0 

4.00 

2.5 

5.66 

3.0 

8.00 

Again  inserting  arithmetical  and  geometrical  means  between  the 
terms  of  the  respective  series  we  have: 


A.  P. 

G.P. 

0.00 

1.00 

0.25 

1.19 

0.50 

1.41 

0.75 

1.69 

1.00 

2.00 

1.25 

2.38 

1.50 

2.83 

1.75 

3.36 

2.00 

4.00 

2.25 

4.76 

By  continuing  this  process  each  consecutive  three  figure  number 
may  finally  be  made  to  appear  in  the  second  column,  so  that,  to 
this  degree  of  accuracy,  the  product  of  any  two  such  numbers 
may  be  found  by  the  process  previously  explained.  The  decimal 
points  of  the  factors  may  be  ignored  in  this  work,  as  for  example, 
the  product  of  2.38  X  14.1  is  the  same  as  that  of  238  X  14.1 


236        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§126  . 

except  in  the  position  of  the  decimal  point.  The  correct  position 
of  the  decimal  point  can  be  determined  by  inanection  after  the 
significant  figures  of  the  product  have  been  obtained.  Using 
the  above  table  we  find  2.38  X  14.1  =  33.6. 

The  above  table,  when  properly  extended,  is  a  table  of  loga- 
rithms. As  geometrical  and  arithmetical  progressions  different 
from  those  given  above  might  havo  been  used,  the  number  of 
possible  systems  of  logarithms  is  indefinitely  great.  The  first 
column  of  figures  contains  the  logarithms  of  the  numbers  that 
stand  opposite  them  in  the  second  column.  Napier,  by  this 
process,  said  he  divided  the  ratio  of  1.00  to  2.00  into  "100  equal 
ratios,"  by  which  he  referred  to  the  insertion  of  100  geometrical 
means  between  1.00  and  2.00.  The  "number  of  the  ratio"  he 
called  the  logarithm  of  the  number,  for  example,  0.75  opposite 
1.69,  is  the  logarithm  of  1.69.  The  word  logarithm  is  from  two 
Greek  words  meaning  "  The  number  of  the  ratios."  In  order  to 
produce  a  table  of  logarithms  it  was  merely  necessary  to  compute 
numerous  geometrical  means;  that  is,  no  operations  except  multi- 
plication and  the  extraction  of  square  roots  were  required.  But 
the  numerical  work  was  carried  out  by  Napier  to  so  many  decimal 
places  that  the  computation  was  exceedingly  difficult. 

The  news  of  the  remarkable  invention  of  logarithms  induced 
Henry  Briggs,  professor  at  Gresham  College,  London,  to  visit 
Napier  in  1615.  It  was  on  this  visit  that  Briggs  suggested  the  ad- 
vantages of  a  system  of  logarithms  ia  which  the  logarithm  of 
10  should  be  1,  for  then  it  would  only  be  necessary  to  insert  a 
sufficient  number  of  geometrical  means  between  1  and  10  to 
get  the  logarithm  of  any  desired  number.  With  the  encourage- 
ment of  Napier,  Briggs  undertook  the  computation,  and  in  1617, 
published  the  logarithms  of  numbers  from  1  to  1000  and,  in 
1624,  the  logarithms'of  numbers  from  1  to  20,000,  and  from  90,000 
to  100,000  to  fourteen  decimal  places.  The  gap  between  20,000 
and  90,000  was  filled  by  a  Hollander,  Adrian  Vlacq,  whose  table, 
published  in  1628,  is  the  source  from  which  nearly  all  the  tables 
since  published  have  been  derived. 

126.  Graphical  Computation  of  the  Terms  of  a  Geometrical 
Progression.  Draw  the  lines  y  =  x  and  y  =  rx,  Fig.  100.  From 
the  point  (1,  r)  on  ?/  =  rx  draw  a  horizontal  line  io  y  =  x,  thence 


§127]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  237 

a  vertical  line  toy  =  rx,  etc.,  thereby  forming  the  "stairway"  of 
line  segments  between  y  =  x  and  y  =  rx  a,s  shown  in  Fig.  100. 
Then  the  points,  N,  P,  Q,  etc.,  have  the  ordinates  r,  r'\  r^,  etc., 
as  required,  for,  to  obtain  the  ordinate  of  P,  or  PD,  the  value  of  x 
used  was  OD  =  r,  hence  P  is  the  point  on  y  =  rx  for  a;  =  r,  or 


y 


PD 


Likewise  Q  is  by  construction  the  point  on  y 


rx  for  X  =  r^,  hence  the  y  of  the  point  Q  =  r  X  r'  =  r^,  etc. 


8 

T 
6 

Y 

U 

/  h 

/  1 

4 

q/ 

:/i 

7 

3 

./ 

'// 

/• 

2 

N 

/ 

^ 

/ 

>r' 

M 

/ 

>■>■- 

---' 

■  r 

r— " 

4— ,- 



]rr- 

r^ 

J 

c 

a-2-1012345 

Fig.  100.^ — Graphical  construction  of  the  successive  terms  of  a  G.  P. 
In  the  diagram  r  =3/2,  and  the  curve  isy  =  (3/2)*.  / 

The  points  P,  Q,  etc.,  are  now  carried  horizontally  to  points 
whose  abscissas  are,  respectively,  2,  3,  4,  etc.,  thus  giving  points 
on  the  curve  for  y  =  r*. 

127.  Graphical  Computation  of  Logarithms.  In  Fig.  100  the 
termis  of  a  geometrical  progression  of  first  term  1  and  ratio  IN  =  r 
are  represented  as  ordinates  arranged  at  equal  intervals  along  OX. 
Fig.  100  is  drawn  to  scale  for  the  value  of  r  =  1.5.  Fig.  101  is 
a  similar  figure  drawn  for  r  =  2,  in  which  a  process  is  used  for 
locating  intermediate  points  of  the  curve,  so  that  the  locus  may 


238        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§127 

be  sketched  with  greater  accuracy.  The  lines  y  =•  x  and  y  =  rx 
(in  this  case  y  =  2x)  are  drawn,  and  the  "stairway"  constructed 
as  before  (See  §126).  Vertical  lines  drawn  through  a;  =  —2,  —1, 
0,  1,  2,  3,  .  .  .  and  horizontal  lines  drawn  through  the  hori- 
zontal tread  of  each  step  of  the  stairWay  divides  the  plane  into  a 
large  number  of  rectangles.  Starting  at  M  and  sketching  the 
diagonals  of  successive  cornering  rectangles  the  smooth  curve 


-.      -1         0        1        2        3        -        . 
Fig.  101. — Graphical  construction  of  the  curve  y  =  2". 

MNP  is  obtained.  Intermediate  points  of  the  curve  are  located  by 
doubling  the  number  of  vertical  lines  by  bisecting  the  distances 
between  each  original  pair,  and  then  by  increasing  the  number  of 
horizontal  lines  in  the  following  manner:  Draw  the  line  y  =  s/r  x 
(in  the  case  of  the  Fig.  101,  y  =  V'2  x).  At  the  points  where 
this  line  cuts  the  vertical  risers  of  each  step  of  the  "stairway" 
(some  of  these  points  are  marked  .A,  B,  C  in  the  diagram)  draw  a 


§127]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  239 

new  set  of  horizontal  lines.  Each  of  the  original  rectangles  is  thus 
divided  into  four  smaller  rectangles.  Starting  at  M  and  sketching 
a  smooth  curve  along  the  diagonals  of  successive  cornering  rec- 
tangles, the  desired  graph  is  obtained.  By  the  use  of  the  straight 
line  y  =  -s/r  x  another  set  of  intermediate  points  may  be  located, 
and  so  on,  and  the  resulting  curve  thus  drawn  to  any  degree  of 
accuracy  required.  In  explaining  this  process,  the  student  will 
show  that  the  method  of  construction  just  used  consists  in  the 
doubling  of  the  number  of  horizontal  lines  of  the  figure  by  the 
successive  insertion  of  geometrical  means  between  the  terms  of  a 
geometrical  progression,  while  at  the  same  time  the  number  of 
vertical  lines  is  successively  doubled  by  the  insertion  of  arithmet- 
ical means  between  the  terms  of  an  arithmetical  series.  Thus  the 
graphical  work  of  construction  of  the  curve  corresponds  to  the 
successive  insertion  of  geometrical  and  arithmetical  means  in  the 
two  series  discussed  in  §125. 

As  explained  above,  the  ordinate  y  of  any  point  of  the  curve 
MNP  of  Fig.  101  is  a  term  of  a  geometrical  progression,  and  the 
abscissa  x  of  the  same  point  is  the  corresponding  term  of  an 
arithmetical  progression.  Since,  when  y  is  given,  the  value  of  x 
is  determined,  we  say,  by  definition,  that  a;  is  a  function  of  y 
(§6).  This  particular  functional  relation  is  so  important  that 
it  is  given  a  special  name:  x  is  called  the  logarithm  of  y,  and  the 
statement  is  abbreviated  by  writing 

X  =  logy, 
but  to  distinguish  from  the  case  in  which  some  other  geometrical 
progression  might  have  been  used,  the  ratio  of  the  progression 
may  be  written  as  a  subscript,  thus 

X  =  logr?/, 
which  is  read  "x  is  the  logarithm  of  y  to  the  base  r." 

The  ratio  of  the  geometrical  progression,  or  r,  is  called  the  base. 

If  we  assume  that  the  process  of  locating  the  successive  sets  of 
intermediate  points  by  the  construction  of  successive  geometrical 
means  will  lead,  if  continued  indefinitely,  to  the  generation  of 
the  curve  MNP  without  breaks  or  gaps,  then  we  may  say  that  in 
the  equation 

X  =  lOgry,  ;  (1) 


240        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§127 

the  logarithm  is  a  function  of  y  defined  for  all  -positive  values  of  y 
and  for  all  halites  of  x. 

It  is  seen  at  once  from  the  method  of  construction  used  in  Fig. 
101  that  the  values  of  y  at  a;  =  1,  2,  3,  4,  ...  ,  are  respectively 
y  =  r,  T^,  r-',  r*,  .  .  ,  and  the  values  oi  y  a.t  x  =  1/2,  3/2,  5/2, 
.  .  .  ,  are  y  =  r^,  r^,  r^,  .  .  .  ,  respectively,  and  similarly  for 
other  intermediate  values  of  x.  In  other  words,  the  equation 
connecting  the  two  variables  x  and  y  may  be  written 

y  =  r^  (2) 

Thus,  when  the  values  of  a  variable  x  run  over  an  arithmetical 
progression  {of  first  term  0)  while  the  corresponding  values  of  a 
variable  y  run  over  a  geometrical  progression  {of  first  term  1),  the 
relation  between  the  variables  may  be  written  in  either  of  the  forms 
(1)  or  (2)  above.  Equation  (2)  is  called  an  exponential  equation 
and  y  is  said  to  be  an  exponential  function  of  x,  while  in  (1)  x 
is  said  to  be  a  logarithmic  function  of  y.  The  student  has  fre- 
quently been  called  upon  in  mathematics  to  express  relations 
between  variables  in  two  different  or  "inverse"  forms,  analogous 
to  the  two  forms  y  =  r'  and  x  =  logr?/.  For  example,  he  has 
written  either 

y  =  x^,  01  X  =  ±  y/y; 


and  either 


y  =  X  ^   01  X  =  y^ 


The  graph  of  a  function  is  of  course  the  same  whether  the  equation 
be  solved  for  x  or  solved  for  y. 

Exercises 
1.  Write  the  following  equations  in  logarithmic  form: 

(a)  y  =  l(y;  (d)  u  =  5'; 

(b)  y  =  3';  (e)  z  =  o";  1 

(c)  y  =  a^;  if)   u  =  1.1'. 

>  As  a  matter  of  fact,  both  the  arithmetical  and  the  geometrical  methods  given 
above  define  the  function  only  tor  rational  values  of  x;  that  is,  the  only  values  of 
X  that  come  into  view  in  the  process  explained  above  are  whole  numbers  and 
intermediate  rational  fractions  like  2|,  2j,  2f,  2^j,  2j|,  .   .   . 


§128]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   241 

2.  Write  the  following  equations  in  exponential  form : 

(o)  X  =  logio  y;  (d)  u  =  log,  y; 

(b)  X  =  logs  y,  (e)    t  =  logs  z; 

(c)  X  =  logs  y;  (/)  u  =  loga  x. 

3.  Find  the  values  of  the  following : 

(a)  logio  100;  (d)  log2  64; 

(6)  logs  25;  (e)  log,  81; 

(c)  logs  27;  if)  log ^16. 

128.    The    Subtangent    of    the    Exponential    Curve.     The 

student  is  expected  to  construct  the  curves  described  in  the 
following  exercises  by  the  method  of  §127.  The  inch  or  2  cm. 
may  be  adopted  as  the  unit  of  measure;  the  curves  should  be  drawn 
on  plain  paper  within  the  interval  from  x  =  — 2toa;=-j-;2. 

If  tangents'  be  drawn  to  the  curves  at  x  =  —  2,  —  1,  0,  1,  2, 
it  will  be  noted,  as  nearly  as  can  be  determined  by  experiment, 
that  the  several  tangents  to  any  one  curve  cut  the  X-axis  at  the 
same  constant  distance  to  the  left  of  the  ordinate  of  the  point 
of  tangency.  This  distance  is  called  the  subtangent  of  the  curve. 
This  distance  is  greater  than  unity  if  r  =  2  and  less  than  unity 
if  r  =  3.  The  value  of  the  base  for  which  the  subtangent  is  exactly 
unity  is  later  shown  to  be  a  certain  irrational  or  incommensurable 
number,  approximately  2.7183  .  ,  represented  in  mathematics 

1  It  is  not  easy  to  draw  accurately  the  tangent  to  a  curve  at  a  given  point. 

A  number  of  instruments  have  been  designed  to  assist  in  drawing  tangents  to 
curves.  One  of  these,  called  a  "Radiator,"  will  be  found  listed  in  most  catalogs  of 
drawing  instruments.  Another  instrument  consists  of  a  straight  edge  provided 
with  a  vertical  mirror  as  shown  in  Fig.  102.     When  the  straight  edge  is  placed  across 


Fig.  102. — Mirrored  ruler  for  drawing  the  normal  (and  hence  the ' 
tangent)  to  any  curve. 

a  curve  the  reflection  of  the  curve  in  the  mirror  and  the  curve  itself  can  both  bii 
seen  and  usually  the  curve  and  image  meet  to  form  a  cusp  or  angle.  The  straight 
edge  may  be  turned,  however,  until  the  image  forms  a  smooth  continuation  of  the 
given  curve.  In  this  position  the  straight-edge  is  perpendicular  to  the  tangent  and 
the  tangent  can  then  be  accurately  drawn.  See  Gramberg,  Technische  Messungen, 
1911. 

16 


242        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§129 

by  the  letter  e,  and  called  the  Naperian  base.  This  number,  and 
the  number  ir,  are  two  of  the  most  important  and  fundamental 
constants  of  mathematics. 

Exercises 

Draw  the  following  curves  on  plain  paper  using  1  inch  or  2  cm.  as  the 
unit  of  measure;  make  the  tests  referred  to  in  the  second  paragraph  of 
§128. 

1.  Construct  a  curve  similar  to  Fig.  101,  representing  the  equation 
X  =  log2  y,  from  a;=— 2toa;  =  +2,  and  draw  tangents  at  a;  =  —  1, 
X  =  0,  X  =  1,  X  =  2.  > 

2.  Construct  the  curve  whose  equation  is  a;  =  logs  y  from  a;  =  —  2 
to  a;  =  +2,  and  draw  tangents  at  a;  =  —  1,  x  =  0,  a;  =  1,  x  =  2. 

3.  Construct  the  curve  whose  equation  is  x  =  logs.?  y,  and  show  by 
trial  or  experiment  that  the  tangent  to  the  curve  at  x  =  2  cuts  the  X- 
axis  at  nearly  x  =  1,  that  the  tangent  at  x  =  1  cuts  the  X^xis  at 

nearly  x  =  0,  that  the  tangent  at  x  =  0 
cuts  the  X-axis  at  nearly  x  =  —  1,  etc. 

4.  Draw  the  curve  x  =  logo.s  y  and 
show  that  it  is  the  same  as  the  reflection 
of  X  =  log2  y  in  the  mirror  x  =  0. 

Note:  The  student    must    remember 

that    the    experimental    testing    of  the 

properties  of  the  tangents  to  the  curves 

called  for  above  does  not  constitute  mathe- 

FiQ.  103.  matical  proof  of  the  usual  deductive  sort 

famUiar  to  him.     The  experimental  tests 

have  value,   however,  in  preparing   the  student  for  a  rigorous  in- 

V  estigation  of  these  same  properties  when  taken  up  in  the  calculus. 

129.  Slope  of  the  Exponential  Curve.  Let  MP,  Fig.  103,  be 
any  exponential  curve,  y  =  r.  By  the  slope  of  the  curve  at  P 
we  mean  the  slope  of  the  tangent  TP  at  P.  We  have  just  shown 
experimentally  that  the  length  of  the  subtangent  TD  is  constant 
for  all  positions  of  the  point  P  on  the  curve  y  =  r".  We  can 
then  write 

slope  of  curve  at  P  =  ^^^  =  j>  (1) 

1 U        K 

where  k  is  the  constant  length  of  TD.     We  can  also  write 

slope  of  curve  at  P  =  cy,  (2) 

where  c  =  t  • 

k 


§130]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  243 

From  (2)  we  can  conclude  that:  The  slope  of  an  exponential 
curve  at  a  given  point  is  proportional  to  the  ordinate  at  that  point. 

At  the  point  (0,  1)  the  slope  is  c  =  t-     As  we  have  seen,  the 

value  oik  {=  TD)  depends  on  the  value  of  r  in  the  equation  of  the 
curve,  y  =  r".  For  some  values  of  r  it  is  less  than  1,  for  others 
greater  than  1.  We  have  defined  e  as  that  value  of  r  for  which 
k  =  TD  =  1.  This  is  equivalent  to  defining  e  as  that  value 
of  r  for  which  the  curve  y  =  r"  has  the  slope  unity  at  the  point 
(0,  1) .    Later  we  shall  adopt  this  definition  of  e. 

Since  c  =  t  =  1  for  the  curve  y  =  e",  it  follows  from  (2)  that 

for  this  particular  curve  of  the  family  of  curves  y  =  r",  the  slope 
at  any  point  is  equal  to  the  ordinate. 

The  reasoning  of  this  section  is  based  on  the  experimentally  de- 
termined result  that  for  a  given  exponential  curve  the  subtangent 
is  of  constant  length. 

130.*  The  Exponential  Function.  The  expression  a",  where  a 
is  any  positive  number  except  1,  has  a  definite  meaning  and 
value  for  all  positive  or  negative  rational  values  of  x,  for  the 
meaning  of  numbers  affected  by  positive  or  negative  fractional 
exponents  has  been  fully  explained  in  elementary  algebra.  The 
process  outlined  above  likewise  defines  logrS  for  aU  rational 
values  of  x,  but  not  for  irrational  values  of  x,  such  as  -\/2,  VB,  etc. 
As  a  matter  of  fact  the  expression  a'  has,  as  yet,  no  meaning 

assigned  to  it  for  irrational  values  of  x;  thus  10^^  has  no  meaning 
by  the  definitions  of  exponents  previously  given,  for  \/2  is  not  a 
whole  number,  hence  10"^^  does  not  mean  that  10  is  repeated 
as  a  factor  a  certain  number  of  times;  also  \/2  is  not  a  fraction, 
so  that  10"^^  cannot  mean  a  power  of  a  root  of  10.  But  if  any 
one  of  the  numbers  of  the  following  sequence : 

1,     1.4,     1.41,     1.414,     1.4142,     1.41421,   .    . 

be  used  as  the  exponent  of  10,  the  resulting  power  can  be  com- 
puted to  any  desired  number  of  decimal  places.  For  example, 
IQi"  is  the  141th  power  of  the  100th  root  of  10;  to  find  the  100th 
root  we  may  take  the  square  root  of  10,  find  the  square  root  of 


244        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§131 

this  result,  then  find  its  5th  root,  finally  finding  the  5th  root 
of  this  last  result. 

If  the  various  powers  be  thus  computed  to  seven  places  we  find: 

10'*  =25.11887 

10'"  =25.70396 

101" «  =25.94179 

101.4142  =25.95374 

101.41421  =  25.95434  . 

101.414213  =  25.95452  .    .    . 

101.4142135  =  25.95455  . 

Now  the  sequence  of  exponents  used  in  the  first  column  is 
found  by  extracting  the  square  root  of  2  to  successive  decimal 
places.  The  sequence  in  the  second  column  approaches  a  limit. 
This  limit  is  taken  hy  definition  as  the  value  of  10'^''. 

In  general,  if  x  is  an  irrational  number,  a"  is  defined  as  the  limit 
of  a  sequence  of  numbers,  o^',  a'^',  .  .  . ,  a^.  .  . ,  the  exponents 
xi,  xi,  .  .,  Xn,.  .  .  being  a  sequence  of  rational  mumbers 
approaching  a;  as  a  limit. 

It  thus  appears  that  if  a  and  y  are  any  given  positive  numbers, 
there  is  a  number  x,  rational  or  irrational,  which  satisfies  the  equa- 
tion a'  =  y.  The  expression  a'  is  called  the  exponential  func- 
tion of  X  with  "base  a. 

131.  Definitions. — In  the  exponential  equation  a'  =  y: 

The  number  a  is  called  the  base. 

The  number  y  is  called  the  exponential  function  of  x  to  the  base 
a,  and  is  sometimes  written  y  =  expoX. 

The  number  x  is  called  the  logarithm  of  y  to  the  base  ,a,  and 
is  written  x  =  logay-    Thus  in  the  equation  a"  =  y,  x  may  be 
called  either  the  exponent  of  a  or  the  logarithm  of  y. 
The  two  equations, 

y  =  a' 
X  =  logay, 

express  exactly  the  same  relations  between  x  and  y;  one  equation 
is  solved  for  x,  the  other  is  solved  for  y.  The  graphs  are  identical, 
just  as  the  graphs  oi  y  =  x^  and  x  =  ±  \/y  are  identical. 

132.  Common   Logarithms.    In  the  equation   10"°  =  y,  x  is 


§133]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  245 

called  the  common  logarithm  of  y.  It  is  also  called  the  Brigg's 
logarithm  of  y.  Thus,  the  common  logarithm  of  any  number  is 
the  exponent  of  the  power  to  which  10  must  be  raised  to  produce 
the  given  number.  Thus  2  is  the  common  logarithm  of  100, 
since  10"  =  100;  likewise  1.3010  will  be  found  to  be  the  con^mon 
logarithm  of  20  correct  to  4  decimal  places,  since  lO^-'""  =  20.00. 

133.  Systems  of  Logarithms.  If  in  the  exponential  equation 
y  =  a',  where  a  is  any  positive  number  except  1,  different  values 
be  assigned  to  y  and  the  corresponding  values  of  x  be  computed 
and  tabulated,  the  results  constitute  a  system  of  logarithms. 
The  number  of  different  possible  systems  is  unlimited,  as  abeady 
noted  in  §125.  -As  a  matter  of  fact,  however,  only  two  systems 
have  been  computed  and  tubulated;  the  natural,  or  Naperian,  or 
hyperboUc,  system,  whose  base  is  the  incommensurable  number  e, 
approximately  2.7182818,  and  the  common,  or  Brigg's,  system, 
whose  base  is  10.  The  letter  e  is  set  aside  in  mathematics  to 
stand  for  the  base  of  the  natural  system. 

Natural  logarithms  of  all  numbers  from  1  to  20,000  have 
been  computed  to  17  decimal  places.  The  common  logarithms 
are  usually  printed  in  tables  of  4,  5,  6,  7  or  8  decimal  places. 

It  will  be  found  later  that  the  graphs  of  all  logarithmic  functions 
of  the  form  x  =  logo  y  can  be  made  by  stretching  or  by  contract- 
ing in  the  same  fixed  ratio  the  ordinates  of  any  one  of  the  logarith- 
mic curves.  That  is,  the  logarithms  of  one  system  can  be  ob- 
tained from  those  of  another  system  by  multiplying  by  a  constant 
factor.  For  this  reason  numerical  tables  in  more  than  one 
system  of  logarithms  are  unnecessary. 

In  the  following  pages  the  common  logarithm  of  any  number  n 
wiU  be  written  log  n,  and  not  logu  n;  that  is,  the  base  is  supposed 
to  be  10  unless  otherwise  designated;  In  x  for  logeS  and  Ig  x  for 
logic  X  are  also  used. 

Exercises 

Write  the  following  in  logarithmic  notation: 

1.  103      =  1000.  6.  e"  =  y. 

2.  10-3    ^  0.001.  7.  10»"     =  1.7783. 

3.  10»       =  1.  8.  lO»Mio  =  2. 

4.  IP      =  121.  9.  oi  =  a. 

5.  16«"  =  2.  10.  10i°8io!'  =  y. 
Express  the  following  in  exponential  notation : 


246        Ea^EMENTARY  MATHEMATICAL  ANALYSIS      [§134 


11.  logu  4        =      0.6021.  16.  log-^^iOO  =        I 

12.  log  10000   =       4.  17.  logj7(l)     =  -ll. 

13.  log  0.0001  =  -  4.  18.  logic  10   =       i- 

14.  logs  1024    =     10.  19.  log  1         =      [O. 
16.  log.  o          =       1.  20.  logal        =      [o. 

134.  Graphical  Table. 

function  defined  by  the  two  progressions  whose  use  was  suggested 


In  Fig.  104  is  shown  the  graph  of  the 


10 

N 

/ 

/ 

/ 

/ 

/ 

/ 

L 

-i 

^f 

10 

N 

/ 

or 

/ 

A 

r  = 

10 

/ 

/ 

y 

/ 

/ 

/ 

y" 

y 

^ 

^ 



^ 

1 

0 

0 

1 

0 

2 

0 

3 

0 

4 

0 

5 

0 

6 

0 

7 

0 

8_ 

0 

9fl 

[o 

Fig.  104. — The  curve  L  =  logioiV. 

by  Briggs  to  Napier,  and  which  are  referred  to  in  the  last  para- 
graph of  §125.  By  inserting  means  three  times  between  0 
and  1  in  the  arithmetical  progression  and  between  1  and  10  in  the 
geometrical  progression,  we  get- 


A.  P.  or 

logarithms 

G.  P.  or 

numbers 

Exponential 
form  of  G.  P. 

0.000 

1.000 

lOO-oO" 

0.125 

1.334 

X00.126 

0.250 

1.778 

100.260 

0.375 

2.371 

100.375 

0.500 

3.162 

IQO.eoo 

0.625 

4.217 

100. 6S6 

0.750 

5.623 

100.760 

0.875 

7.499 

100.»76 

1.000 

10.000 

IQi.ooa 

§135]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  247 

If  we  let  L  stand  for  the  logarithm  of  the  number  N,  the 
functional  relation  is  obviously  L  =  logioiV,  or  iV  =  10^.  The 
curve,  Fig.  104,  may  now  be  used  as  a  graphical  table  of  logarithms 
from  which  the  results  can  be  read  to  about  2  decimal  places. 
The  logarithms  of  numbers  between  1  and  10  may  be  read  directly 
from  the  graph.  Thus,  logio  7.24  =  0.860.  If  the  logarithm  is 
between  0  and  1,  the  number  is  read  directly  from  the  graph. 
Thus  if  the  logarithm  is  0.273,  the  number  is  1.87. 

If  we  multiply  the  readings  of  the  A/^-scale  by  10",  we  must  add 
n  to  the  readings  on  the  L-scale,  for  lO^A''  =  10^+". 

If  we  divide  the  readings  on  the  A''-scale  by  10",  we  must 
subtract  n  from  the  readings  on  the  L-scale,  for  N/10"  =  10^  ~". 

This  fact  enables  us  to  read  the  logarithms  of  all  numbers  from 
the  graph,  and  conversely  to  find  the  number  corresponding  to 
any  logarithm.  Thus  we  have,  log  72.4  =  1.860,  log  724  = 
2.860,  log  0.724  =  0.860  -  1,  log  0.0724  =  0.860  -  2. 

If  the  logarithm  is  1.273,  the  number  is  18.7. 

If'the  logarithm  is  2.273,  the  number  is  187. 

If  the  logarithm  is  0.273  -  1,  the  number  is  0.187. 

If  the  logarithm  is  0.273  -  2,  the  number  is  0.0187. 

We  observe  that  the  computation  of  a  three  place  table  -oi 
logarithms  would  not  involve  a  large  amount  of  work.  Such  a 
table  has  actually  been  computed  in  drawing  the  curve  of  Fig. 
104.  The  original  tables  of  Briggs  and  Vlacq  involved  an  enor- 
mous expenditure  of  labor  and  extraordinary  skill,  or  even  genius 
in  computation,  because  the  results  were  given  to  fourteen  places 
of  decimals. 

135.  Properties  of  Logarithms.  The  following  properties  of 
logarithms  follow  at  once  from  the  general  properties  or  laws  of 
exponents. 

(1)  The  logarithm  of  1  is  0  in  all  systems.  For  a"  =  1,  that 
is,  logal  =  0.  In  Fig.  101,  note  that  the  curve  passes  through 
(0,  1). 

(2)  The  logarithm  of  the  base  itself  in  any  system  is  1.  For 
a^  =  o,  that  is,  log„a  =  1.  In  Fig.  101,  by  construction  N  is 
always  the  point  (1,  r),  where  r  is  the  ratio  of  the  first  or  funda- 
mental progression  in  which  means  are  inserted;  in  the  present 
notation,  this  is  the  point  (1,  a). 


248        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§136 

(3)  Negative  numbers  have  no  logarithms.  This  follows  at 
once  from  Fig.  101.  In  Figs.  100,  101,  and  104,  note  that  the 
curves  do  iiot  extend  below  the  X-axis. 

Note:  While  negative  numbers  have  no  logarithms,  this  does  not 
prevent  the  computation  of  expressions  containing  negative  factors 
and  divisors.  Thus  to  compute  (287)  X  (-  374),  find  (287)  X  (374) 
by  logarithms  and  give  proper  sign  to  the  result. 

136.  Logarithm  of  a  Product.  Let  n  and  r  be  any  two  positive 
numbers,  and  let 

logo  n  =  X  and  logo  r  =  y.  (1) 

Then,  by  definition  of  a  logarithm,  §131, 

n  —  a'  and  r  =  a".  (2) 

Hence, 

nr  =  a"  a"  =  a''*^ 

Therefore,  by  definition  of  a  logarithm,  §131, 

logo  nr  =  X  +  y, 
or,  by  (1), 

log.  nr  =  loga  n  +  log.  r.  (3) 

Hence,  the  logarithm  of  the  product  of  two  numbers  is  equal  to 
the  sum  of  the  logarithms  of  those  numbers. 
In  the  same  way,  if  log.  s  =  «,  then 

nrs  =  a"^-'-', 
that  is, 

log.  nrs  =  log.  n  +  log.  r  +  log.  s. 

Exercises 

Find  the  results  by  the  formulas  and  check  by  the  curve  of  Fig.  104. 

1.  Given  log  2  =  0.3010,  and  log  3  =  0.4771;  find  log  6;  find  log  18. 

2.  Given  log  5  =  0.6990  and  log  7  =  0.8451;  find  log  35. 

3.  Given  log  9  =  0.9542,  find  log  81. 

4.  Given  log  386  =  2.5866  and  log  857  =  2.9330;  find  the  logarithm 
of  their  product. 

6.  Given  log  llx  =  1.888  and  log  11  =  1.0414;  find  log  x. 

137.  Logarithm  of  a  Quotient.  Let  n  and  r  be  any  two  positive 
numbers,  and  let 

log.  n  =  X  and  log.  r  =  y.  (1) 


§138]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  249 

From  (1),  by  the  definition  of  a  logarithm, 

n  =  O'  and  r  =  a". 
Hence, 

n/r  =  a"  -T-  a"  =  a''". 

Therefore,  by  definition  of  a  logarithm, 

loga  {n'/r)  =  X  -  y, 
or  by  (1) 

logo  (n/r)  =  logo  n  -  log,  r.  (2) 

Hence,   the   logarithm  of  the  quotient  of  two  numbers  equals  the 
logarithm  of  the  dividend  minus  the  logarithm  of  the  divisor. 

Exercises 

Find  the  results  by  the  formulas  and  check  by  the  curve  of  Fig.  104. 

1.  Given  log  6  =  0.6990  and  log  2  =  0.3010;  find  log  (5/2);  find 
log  0.4. 

2.  Given  log  63  =  1.7993,  and  log  9  =  0.9542;  find  log  7. 

3.  Given  log  84  =  1.9243  and  log  12  =  1.0792;  find  log  7. 

4.  Given  log   1776  =  3.2494  and   log    1912  =  3.2815;    find  log 
1776/1912;  find  log  1912/1776. 

5.  Given  log  a;/12  =  0.4321  and  log  12  =  1.0792,  find  log  x. 

138.  Logarithm  of  any  Power.    Let  n  be  any  positive  number 
and  let 

logo  n  =  X.  (1) 

From  (1),  by  the  definition  of  a  logarithm, 

n  =  a". 

Raising  both  sides  to  the  pth  power,  where  p  is  any  number  what- 
soever, 

UP  =  a"'. 

Therefore,  by  definition  of  a  logarithm, 

logo  (w)  =  px, 
or,  by  (1), 

logo  (n^)  =  p  logon.  (2) 

Hence,  the  logarithm  of  any  power  of  a  number  equals  the  logarithm 
of  the  number  multiplied  by  the  index  of  the  power. 


250        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§139 

The  above  includes  the  two  cases:  (1)  the  finding  of  the 
logarithm  of  any  integral  power  of  a  number,  since,  in  this  case 
p  is  a  positive  integer;  and  (2)  the  finding  of  the  logarithm  of  any 
root  of  a  number,  since,  in  this  case,  p  is  the  reciprocal  of  the  index 
of  the  root. 

Exercises 

1.  Given  log  2  =  0.3010;  find  log  1024;  find  log  V2;  find  log  y^. 

2.  Given  log  1234  =  3.0913;  find  log  Vl234;  find  log  -^/i^Si. 

3.  Given  log  5  =  0.6990;  find  log  53 ;  find  log  sl. 

4.  Show  that  log  (11/15)  +  log  (490/297)  -  2  log  (7/9)  =  log  2. 
6.  Find  an  expression  for  the  value  of  x  from  the  equation  3'  =  567. 
Solution:  Take  the  logarithm  of  each  side; 

X  log  3  =  log  567. 
But 

log  567  =  log  (3<  X  7)  =  4  log  3  +  log  7. 
Therefore 

X  log  3  =  4  log  3  +  log  7, 
or 

X  =  4  +  (log  7)/(log  3). 

6.  Find  an  expression  for  x  in  the  equation  5'  =  375. 

7.  Given  log  2  =  0.3010  and  log  3  =  0.4771,  find  how  many  digits 
in  6'°. 

8.  Find  an  expression  for  x  from  the  equation 

3»  X  2»+i  =  -v/si^. 

9.  Prove  that  log  (75/16)  -  2  log  (5/9)  +  log  (32/243)  =  log  2. 

139.  Characteristic  and  Mantissa.  The  common  logarithm 
of  a  number  is  always  written  so  that  it  consists  of  a  positive 
decimal  part  and  an  integral  part  which  may  be  either  positive 
or  negative.  Thus  log  0.02  =  log  2  -  log  100  =  0.3010  -  2. 
Log  0.02  is  never  written  —  1.6990. 

When  a  logarithm  of  a  number  is  thus  arranged,  special  names 
are  given  to  each  part.  The  positive  or  negative  integral  part  is 
called  the  characteristic  of  the  logarithm.  The  •positive  decimal 
part  is  called  the  mantissa.  Thus,  in  log  200  =  2.3010,  2  is 
the  characteristic  and  3010  is  the  mantissa.  In  log  0.02  = 
0.3010  —  2,  (—  2)  is  the  characteristic  and  3010  is  the  mantissa. 

Since  log  1=0  and  log  10  =  1,  every  number  lying  between  1 


§139]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  251 

an4  10  has  for  its  common  logarithm  a  number  between  0  and  1 ; 
that  is,  the  characteristic  is  0.  Thus  log  2  =  0.3010,  log  9.99  = 
0.9996,  log  1.91  =  0.2810.     Starting  with  the  equation 

log  1.91  =  0.2810, 
we  have,  by  §136, 

log  19.1  =  log  1.91  +  log  10  =  0.2810  +  1  =  1.2810, 
log  191  =  log  1.91  +  log  100  =  0.2810  +  2  =  2.2810, 
log  1910  =  log  1.91  +  log  1000  =  0.2810  +  3  =  3.2810,  etc. 

Likewise,  by  §137, 

log  0.191  =  log  1.91  -  log  10  =  0.2810  -  1, 
log  0.0191  =  log  1.91  -  log  100  =  0.2810  -  2, 
log  0.00191  =  log  1.91  -  log  1000  =  0.2810  -  3,  etc. 

Since  thexharaoteristic  of  the  common  logarithm  of  any  number 
having  its  first  significant  figure  in  units  place  is  zero,  and  since 
moving  the  decimal  point  to  the  right  or  left  is  equivalent  to 
multiplying  or  dividing  by  a  power  of  10,  or  equivalent  to  adding 
an  integer  to  or  subtracting  an  integer  from  the  logarithm, 
(§134):  (1)  the  value  of  the  characteristic  is  dependent  merely 
upon  the  position  of  the  decimal  point  in  the  number;  (2)  the 
value  of  the  mantissa  is  the  same  for  the  logarithms  of  all 
numbers  that  differ  only  in  the  position  of  the  decimal  point. 
In  particular,  we  derive  therefrom  the  following  rule  for  finding 
the  characteristic  of  the  common  logarithm  of  any  number: 

The  characteristic  of  the  common  logarithm  of  a  number  equals 
the  number  of  places  the  first  significant  figure  of  the  number  is 
removed  from  units'  place,  and  is  positive  if  the  first  significant 
figure  stands  to  the  left  of  units'  place  and  is  negative  if  it  stands 
to  the  right  of  units'  place. 

Thus,  in  log  1910  =  3.2810,  the  first  figure  1  is  three  places  from 
units'  place  and  the  characteristic  is  3.  In  log  0.0191  =  0.2810 
—  2,  the  first  significant  figure  1  is  two  places  to  the  right  of  units' 
place  and  the  characteristic  is  —  2.  A  computer  in  determining 
the  characteristic  of  the  logarithm  of  a  number  first  points  to 
units'  place  and  counts  zero,  then  passes  to  the  next  place  and 
counts  one  and  so  on  until  the  first  significant  figure  is  reached. 


252        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§140 

Logarithms  with  negative  characteristics,  Uke  0.3010  —  1, 
0.3010  —  2,  etc.,  should  be  written  in  the  equivalent  form 
9.3010  -  10,  8.3010  -  10,  etc. 

Exercises , 

1.  What  numbers  have  0  for  the  characteristic  of  their  logarithm? 
What  numbers  have  0  for  the  mantissa  of  their  logarithms? 

2.  Find  the  characteristics  of  the  logarithms  of  the  following  num- 
bers: 1234,  5,  678,  910,  212,  57.45,  345.543,  7,  7.7,  0.7,  0.00000097, 
0.00010097. 

3.  Given  that  log  31,416  =  4.4971,  find  the  logarithms  of  the 
foUowmg  numbers:  314.16,  3.1416,  3,141,600,  0.031416,  0.31416, 
0.00031416. 

4.  Given  that  log  746  =  2.8727,  write  the  numbers  which  have  the 
foUowmg  logarithms:  4.8727,  1.8727,  7.8727  -  10,  9.8727  -  10, 
3.8727,  6.8727  -  10. 

140.  Logarithmic  Tables.  A  table  of  common  logarithms  con- 
tains only  the  mantissas  of  the  logarithms  of  a  certain  convenient 
sequence  of  numbers.  For  example,  a  four  place  table  will  con- 
tain the  mantissas  of  the  logarithms  of  numbers  from  100  to 
1000;  a  five  place  table  will  usually  contain  the  mantissas  of 
the  logarithms  of  numbers  from  1000  to  10,000,  and  so  on.  Of 
course  it  is  unnecessary  to  print  decimal  points  or  characteristics. 

A  table  of  logarithms  should  contain  means  for  readily  obtaining 
the  logarithms  of  numbers  intermediate  to  those  tabulated,  by 
means  of  tabular  differences  and  proportional  parts.' 

The  tabular  differences  are  the  differences  between  successive 
mantissas.  If  any  tabular  difference  be  multiplied  successively 
by  the  numbers  0.1,  0.2,  0.3,  .  .  .  ,  0.8,  0.9,  the  results  are  called 
the  proportional  parts.  Thus,  from  a  four  place  table  we  find 
log  263  =  2.4200.  The  tabular  difference  is  given  in  the  table 
as  16.  If  we  wish  the  logarithm  of  263.7,  the  proportional  part 
0.7  X  16  or  11.2  is  added  to  the  mantissa,  giving,  to  four  places, 
log  263.7  =  2.4211.  This  process  is  known  as  interpolation. 
Corrections  of  this  kind  are  made  with  great  rapidity  after  a 

1  The  student  is  supposed  to  have  Slichter's  Four  Place  Tables,  Macmillan  A 
Co.,  New  York.  The  edition  printed  on  three  sheets  of  heavy  manilla  paper  per- 
forated to  lit  in  notebook  is  preferred.    See  also  tables  at  end  of  this  book. 


§140]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  253 

little  practice.  It  is  obvious  that  the  principle  used  in  the 
correction  is  the  equivalent  of  a  geometrical  assumption  that  the 
graph  of  the  function  is  nearly  straight  between  the  successive 
values  of  the  argument  given  in  the  table.  The  corrections 
should  invariably  be  added  mentally  and  all  the  work  of  interpolation 
should  be  done  mentally  if  the  finding  of  the  proportional  parts  by 
mental  work  does  not  require  multiplication  beyond  the  range  of 
12  X  12.  To  make  interpolations  mentally  is  an  essential  practice, 
if  the  student  is  to  learn  to  compute  by  logarithms  with  any  skiU 
beyond  the  most  rudimentary  requirements. 

A  good  method  is  the  following:  Suppose  log  13.78  is  required. 
First  write  down  the  characteristic  1 ;  then,  with  the  table  at  your 
left,  find  137  in  the  number  column  and  mark  the  corresponding 
mantissa  by  placing  your  thumb  above  it  or  your  first  finger 
below  it.  Do  not  read  this  mantissa,  but  read  the  tabular  differ- 
ence, 32.  From  the  p.  p.  table  find  the  correction,  26,  for  8.  Now 
return  to  the  mantissa  marked  by  your  finger,  and  read  it  increased 
by  26,  i.e.,  1393;  then  place  1393  after  the  characteristic  1 
previously  written  down. 

The  accuracy  required  for  nearly  aU  engineering  computations 
does  not  exceed  3  or  4  significant  figures.  Four  figure  accuracy 
means  that  the  errors  permitted  do  not  exceed  1  percent  of 
1  percent.  Only  a  small  portion  of  the  fundamental  data  of 
science  is  reliable  to  this  degree  of  accuracy.^  The  usual  meas- 
urements of  the  testing  laboratory  fall  far  short  of  it.  Only 
in  certain  work  "in  geodesy,  and  in  a  few  other  special  fields  of 
engineering,  need  more  than  four  place  logarithms  be  used. 

Exercises 

Knd  the  logarithms  of  the  following : 

1.  136.  4.  375.S  7.  2.758. 

2.  752.  5.  217.6  8.  762,700. 

3.  976.  6.  17.62  9.  0.1278. 

^  Fundamental  constants  upon  wMch  much  of  the  calculation  in  applied  science 
must  be  based  are  not  often  known  to  four  figures.  The  mechanical  equivalent  of 
heat  is  hardly  known  to  1  percent.  The  specific  heat  of  superheated  steam  is  even 
less  accurately  known.  The  tensile,  tortional,  and  compressive  strength  of  no 
structural  material  would  be  assumed  to  be  known  to  a  greater  accuracy  than  the 
above-named  constants.  Of  course  no  calculated  result  can  be  more  accurate  than 
the  least  accurate  of  the  measurements  upon  which  it  depends. 


254        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§U1 

141.  Anti-logarithms.  If  we  wish  to  find  the  number  which 
has  a  given  logarithm,  it  is  convenient  to  have  a  table  in  which 
the  logarithm  is  printed  before  the  number.  Such  a  table  is  known 
as  a  table  of  anti-logarithms.  It  is  usually  not  best  to  print 
tables  of  anti-logarithms  to  more  than  four  places;  to  find  a  number 
when  a  five  place  logarithm  is  given,  it  is  preferable  to  use  the 
table  of  logarithms  inversely,  as  the  large  number  of  pages  required 
for  a  table  of  anti-logarithms  is  a  disadvantage  that  is  not  com- 
pensated for  by  the  additional  convenience  of  such  a  table. 

Exercises 

From  a  four  place  table  of  anti-logarithms,  find  the  numbers  cor- 
responding to  the  following  logarithms: 

1.  2.7864.  2.  3.1286.  3.  1.8152. 

4.  9.6278  -  10.  5.  8.1278  -  10.  .   6.  6.1785  -  10. 

142.  Cologarithms.  Any  computation  involving  multiplica- 
tion, division,  evolution,  and  involution  may  be  performed  by 
the  addition  of  a  single  column  of  logarithms.  This  possibility 
is  secured  by  using  the  cologarithms,  instead  of  the  logarithms,  of 
aU  divisors.  The  cologarithm,  or  complementary  logarithm, 
of  a  number  n  is  defined  to  be  (10  —  log  n)  —  10.  The  part 
(10  —  log  n)  can  be  taken  from  the  table  just  as  readily  as  log  n, 
by  subtracting  in  order  all  the  figures  of  the  logarithm,  including  the 
characteristic,  from  9,  except  the  last  figure,  which  must  be  taken 
from  10.  The  subtraction  should,  of  course,  be»done  mentally. 
Thus  log  263  =  2.4200,  whence  colog  263  =  7.5800  -  10.  In 
like  manner  colog  0.0263  =  1.5800.  It  is  obvious  that  the 
addition  of  (10  —  log  n)  —  10  is  the  same  as  the  subtraction  of 
log  n. 

The  convenience  arising  from  this  use  may  be  illustrated  as  follows : 
Suppose  it  is  required  to  find  x  from  the  proportion 

37.42  :x  ::647  :  v'0.S82! 
We  then  have 

2  log  37.4  =  3.1458 
(1/2)  log  0.582  =  9,8825  -  10 
colog  647  =  7.1891  _  10 
log  X  =  0.2174 
X  =  [1. 650]. 


§143]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  255 

It  is  a  good  custom  to  enclose  a  computed  result  in  square 
brackets. 

143.  Arrangement  of  Work.  All  logarithmic  work  should  be 
arranged  in  a  vertical  column  and  should  be  done  with  pen  and 
ink.  Study  the  formula  in  which  numerical  values  are  to  be 
substituted  and  decide  upon  an  arrangement  of  your  work  in  the 
vertical  column  which  will  make  the  additions,  subtractions,  etc., 
of  logarithms  as  systematic  and  easy  as  possible.  Fill  out  the 
vertical  column  with  the  names  and  values  of  the  data  before 
turning  to  the  table  of  logarithms.  This  is  called  blocking  out 
the  work.  The  work  is  not  properly  blocked  out  unless  every 
entry  in  the  work  as  laid  out  is  carefully  labelled,  stating  exactly 
the  name  or  value  of  the  magnitude  whose  logarithm  is  taken, 
and  unless  the  computation  sheet  bears  a  formula  or  statement 
fully  explaining  the  purpose  of  the  work. 

Computation  Sheet,  Form  M7,  is  suitable  for  general  logarithmic 
computation. 

Illustration  1.  Find  the  weight  in  pounds  of  a  circular  disk 
of  steel  of  radius  2.64  feet  and  thickness  0.824  inch,  if  the  specific 
gravity  of  the  steel  be  7.86. 

Formula  :  Call  r  the  radius  in  feet  and  t  the  thickness  in  inches. 
Take  64.48  as  the  weight  of  one  cubic  foot  of  water.  Then  the  weight 
in  pounds  w  is  given  by 


w  =-nrH</12)(64.48)(7.86). 


Work: 


logx 

=  0.4971 

21ogr 

=  0.8432 

log* 

=  9.9159  -  10 

colog  12 

=  8.9208  -  10 

log  64.48 

=  1.8095 

log  7.86 

=  1.8954 

log  w 

=  3.8819 

w 

=  [761.9  lbs.] 

Illustration  2.    Compute  the  value  of  x  if 


■^  I  O.K 
X  ==yj 


1673  X  2.142_ 
3.871 


256        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§143 

Wobk: 

log  0.1673  =9.2235-10 

log  2,142  =0.3308 

colog  3 . 871  =  9.4122  -  10 

Sum  =  8.9665  -  10 

log  a;  =9.6555-10 

X  =  [0.4524] 

Remark  1.  In  writing  a  decimal  fraction  without  an  integral  part, 
always  place  a  zero  before  the  decimal  point;  thus  0. 1673. 

Remark  2.  Note  the  order  of  logarithmic  work:  First,  write  the 
formula;  next  block  out  the  work  by  writing  down  the  first  column,  as 
in  illustrations  above;  finally  hunt  up  logarithms  from  table  and  place 
in  second  column  of  work. 

Remark  3.  After  addition  note  that  23.8819  —  20  is  written  as 
3.8819. 

Remark  4.  In  dividing  a  logarithm  like  8.9665  —  10  by  3,  first 
call  the  expression  28.9665  —  30  and  then  divide  by  3.  If  division 
by  5  had  been  required,  the  dividend  would  of  course  have  been  called 
48.9665  -  50. 

Exercises 

1.  Compute  by  logarithms  the  value  of  the  following :  2.56  X  3.11 
X  421;  7.04  X  0.21  X  0.0646;  3215  X  12.82  -^  864. 

2.  Compute  the  following  by  logarithms:  81'  ■^  17«;  158\/0^; 
(343/892)';  Vl893  \/l912/446='. 

3.  Compute  the  following  by  logarithms:  (2.7182)'"';  (7.41)-*; 
(8.31)«-". 

4.  Solve  the  following  equations:  5*  =  10;  3*-^  =  4;  log»  71  = 
1.21. 

5.  Find  the  amount  of  $550  for  fifteen  years  at  5  percent  compound 
interest. 

6.  A  corporation  is  to  repay  a  loan  of  $200,000  by  twenty  equal 
annual  payments.  How  much  will  have  to  be  paid  each  year,  if 
money  be  supposed  to  be  worth  5  percent? 

Let  X  be  the  amount  paid  each  year.  As  the  debt  of  $200,000  is 
owed  now,  the  present  value  of  the  twenty  equal  payments  of  x  dollars 
each  must  add  up  to  the  debt  or  $200,000.  The  sum  of  x  dollars 
to  be  paid  n  years  hence  has  a  present  worth  of  only 

X 

(1.05)" 


§143]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  267 

if  money  be  worth  6  percent  compound  interest.  The  present  value, 
then,  of  X  dollars  paid  one  year  hence,  x  dollars  paid  two  years  hence, 
and  so  on,  is 


1.05       (1.05)!'    '   (1.05)3    ,    .    .    .     ,    (105)20 

This  is  a  geometrical  progression  which  can  be  summed  by  the  usual 
formula. 

The  result  in  this  case  is  the  value  of  an  annuity  payable  at  the  end 
of  each  year  for  twenty  years  that  a  present  payment  of  $200,000  will 
purchase.  Four  place  tables  will  not  give  more  than  3  place  accuracy 
in  this  and  the  following  problem.  To  get  4  or  5  place  accuracy, 
6  place  tables  would  be  required. 

7.  It  is  estimated  that  a  certain  power  plant  costing  $220,000  will 
become  entirely  worthless  except  for  a  scrap  value  of  $20,000  at  the 
end  of  twenty  years.  What  annual  sum  must  be  set  aside  to  amount 
to  the  cost  of  replacement  at  the  end  of  twenty  years,  if  5  percent 
compound  interest  is  realized  on  the  money  in  the  depreciation  fund? 

Let  the  annual  amount  set  aside  be  x.  In  this  case  the  twenty 
equal  payments  are  to  have  a  value  of  $200,000  twenty  years  hence, 
while  in  the  preceding  problem  the  payments  were  to  be  worth 
$200,000  now.     In  this  case,  therefore, 

a;(1.05)"  +  a;(1.05)"  +  x(1.05)"  +  .    .    . 

+  x(1.05)2  +  x(1.05)  +  s  =  $200,000. 

The  geometrical  progression  is  to  be  summed  and  the  resulting 
equation  solved  for  x. 

8.  The  population  of  the  United  States  in  1790  was  3,930,000  and 
in  1910  it  was  93,400,000.  What  was  the  average  rate  percent  in- 
crease for  each  decade  of  this  period,  assuming  that  the  population 
increased  in  geometrical  progression  with  a  uniform  ratio  for  the  entire 
period. 

9.  Find  the  surface  and  the  volume  of  a  sphere  whose  radius  is  7.211. 

10.  Find  the  weight  of  a  cone  of  altitude  9.64  inches,  the  radius  of 
the  base  being  5.35  inches,  if  the  cone  is  made  of  steel  of  specific 
gravity  7.93. 

11.  Find  the  weight  of  a  sphere  of  cast  iron  14.2  inches  in  diameter, 
if  the  specific  gravity  of  the  iron  be  7.30. 

12.  In  twenty-four  hours  of  continuous  pumping,  a  pump  dis- 
charges 450  gallons  per  minute;  by  how  much  will  it  raise  the  level  of 
water  in  a  reservoir  having  a  surface  of  1  acre?  (1  acre  =  43560  sq.ft.) 

17 


258        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§144 

144.  Trigonometric  Computations.  Logarithms  of  the  trigo- 
nometric functions  are  used  for  computing  the  numerical  value 
of  expressions  containing  trigonometric  functions,  and  in  the 
solution  of  triangles.  Right  triangles,  previously  solved  by 
use  of  the  natural  functions,  are  often  more  readily  solved  by 
means  of  logarithms.  (See  §62.)  The  tables  of  trigonometric 
functions  "contain  adequate  explanation  of  their  use,  so  that 
detailed  instructions  need  not  be  given  in  this  place.  Two  new 
matters  Of  great  importance  are  met  with  in  the  use  of  the  loga-' 
rithms  of  the  trigonometric  functions  that  do  not  arise  in  the  use 
of  a  table  of  logarithms  of  numbers,  which,  on  that  account,  re- 
quire especial  attention  from  the  student: 

(1)  In  interpolating  in  a  table  of  logarithms  of  trigonometric 
functions,  the  corrections  to  the  logarithms  of  all  co-functions  must 
be  svhtracted  and  not  added.  Failure  to  do  this  is  the  cause  of 
most  of  the  errors  made  by  the  beginner. 

(2)  To  secure  proper  relative  accuracy  in  computation,  the 
S  and  T  functions  must  be  used  in  interpolating  for  the  sine  and 
tangent  of  small  angles. 

In  the  following  work,  four  place  tables  of  logarithms  are 
supposed  to  be  in  the  hands  of  the  students. 

Exercises 

1.  A  lateral  face  of  a  right  prism,  whose  base  is  a  square  17.45  feet 
on  a  side,  is  cut  in  a  line  parallel  to  the  base  by  a  plane  making  an 
angle  of  27°  15'  with  the  face.  Find  the  area  of  the  section  of  the 
prism  made  by  the  cutting  plane. 

2.  The  perimeter  of  a  regular  decagon  is  24  feet.  Find  the  area  of 
the  decagon. 

3.  To  find  the  distance  between  two  points  B  and  C  on  opposite 
banks  of  a  river,  a  distance  CA  is  measured  300  feet,  perpendicular 
to  CB.  At  A  the  angle  CAB  is  found  to  be  47°  27'.  Find  the 
distance  CB. 

4.  In  running  a  line  18  miles  in  a  direction  north,  2°  13.2'  east, 
how  far  in  feet  does  one  depart  from  a  north  and  south  line  passing 
through  the  place  of  beginning? 

5.  How  far  is  Madison,  Wisconsin,  latitude  43°  5',  from  the  earth's 
axis  of  rotation,  assuming  that  the  earth  is  a  sphere  of  radius  3960 
miles? 


§144]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  259 

6.  A  man  walking  east  7°  15'  north  along  a  river  notices  that  after 
passing  opposite  a  tree  across  the  river  he  walks  107  paces  before  he 
is  in  line  with  the  shadow  of  the  tree.  Time  of  day,  noon.  How  far 
is  it  across  the  river? 

7.  Solve  the  right-angled  triangle  in  which  one  leg  =  2\/3  and  the 
hypotenuse  =  2ir. 

8.  The  moon's  radius  is  1081  miles.  When  nearest  the  earth,  the 
moon's  apparent  diameter  (the  angle  subtended  by  the  moon's  disk  as 
seen  from  the  position  of  the  earth's  center)  is  32'.79.  When  farthest 
from  the  earth,  her  apparent  diameter  is  only  28'.  73.  Find  the 
nearest  and  farthest  distances  of  the  moon  in  miles. 

9.  A  pendulum  39  inches  long  vibrates  3°  5'  each  side  of  its  mean 
position.  At  the  end  of  each  swing,  how  far  is  the  pendulum  bob 
above  its  lowest  position? 

10.  If  the  deviation  of  the  compass  be  2°  1.14'  east,  how  many  feet 
does  magnetic  north  depart  from  true  north  in  a  distance  of  1  mile 
true  north? 

11.  Solve  

X  :  1.72  =  427  :  V2gh, 

a  g  =  32.2  and  h  =  78.2. 

12.  A  substance  containing  20  percent  of  impurities  is  to  be  purified 
by  crystallization  from  a  mother  liquid.  Each  crystallization  reduces 
the  impurity  88.6  percent.  How  many  crystallizations  will  produce 
a  substance  0.9999  pure? 

13.  Compute  the  value  of  (1  —  ae"'")"  where  a  =  15.6,  b  =  -t~' 

\ 

X  =  10,  71  =  2,  2/  =  2.5. 

14.  Find  the  volume  of  a  cone  if  the  angle  at  the  apex  be  15°  38' 
and  the  altitude  17.48  inches. 

15.  The  angle  subtended  by  the  sun's  diameter  as  seen  from  the 
earth  is  32'.06.  Find  the  diameter  of  the  sun  in  miles,  if  the  distance 
from  the  earth  to  the  sun  be  92.8  million  miles. 

16.  Compute  by  logarithms  four  values  of  p  from  the  equation 
V  =  32.2(ii."8,  for  d  =  2,3,  4,  5. 

17.  Solve  3'  =  405  for  the  value  of  x. 

18.  Compute: 

23.07  X  0.1354  X  -s/234 
13.54 

What  advantage  is  there  in  using  the  co-logarithm  of  the  denomi- 
nator? 


260        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§145 


145.  *  Logarithmic  and  Exponential  Curves.  The  graphical 
construction  of  the  exponential  curve  has  already  been  explained. 
It  was  noted  that  curves  whose  equations  are  of  the  form  y  =  r^ 
pass  through  the  point  (0,  1),  and  that  the  slope  of  the  curves 
for  positive  values  of  x  is  steeper  the  larger  the  value  selected  for 
the  number  r.  In  a  system  of  exponential  curves  y  =  r'  passing 
through  the  point  (0,  1),  or  the  point  M  of  Fig.  105,  we  have 
assumed  (§129)  that  there  is  one  curve  passing  through  M 
with  slope  1.    The  equation  of  this  particular  curve  we  have 

called  y  =  e',  thereby  de- 
fining the  number  e  as  that 
value  of  r  for  which  the 
curve  y  =  r'  passes  through 
the  point  (0, 1)  with  slope  1. 
In  §130  there  was  de- 
veloped on  the  basis  of  the 
first  definition  of  e,  the 
characteristic  property  of 
the  curve  2/  =  e":  The  slope 
of  the  curve  y  =  e'  at  any 
point  is  equal  to  the  ordinate 
of  that  point.  This  fact, 
developed  experimenrtaUy 
in  §129,  will  now  be  shown 
to  foUow  necessarily  from 
the  definition  of  e  just 
given. 

Select  the  point  P  on  the 
curve  y  =  e'  at  any  point 
desired.  Draw  a  line  through  P  cutting  the  curve  at  any  neigh- 
boring point  Q.  (Fig.  105.)  A  line  like  PQ  that  cuts  a  curve  at 
two  points  is  called  a  secant  line.  As  the  point  Q  is  taken  nearer 
and  nearer  to  the  point  P  (P  remaining  fixed),  the  limiting  position 
approached  by  the  secant  PQ  is  called  the  tangent  to  the  curve 
at  the  point  P.  This  is  the  general  definition  of  the  tangent  to 
any  curve. 

The  slope  of  the  secant  joining  P  to  the  neighboring  point  Q 
is  HQ/PH.    As  the  point  Q  approaches  P  this  ratio  approaches 


Fig.  105.- 


-Definition  of  tangent  to  a 
curve. 


§145]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  261 

the  slope  of  the  tangent  to  y  =  e'  at  the  point  P-     Let  OD  =  x 

and  PH  =  h;  then  OE  =  x  +  h,  also  DP  =  e'  and  EQ  =  e'+K 

Since  HQ  is  the  y  of  the  point  Q  minus  the  y  of  the  point  P,  we 

have 

jjQ  ^gx+h  —  gx^    ^e'^  —  1 

Pff~       ^         ~  ^'      /i 

Now  the  slope  of  j/  =  e*  at  P  is  the  limit  of  the  above  expression 
as  Q  approaches  P,  or  as  h  approaches  zero.     That  is 

slope  of  e'  at  P  =  J^'J  e'  [^^]  •  (D 

As  the  point  Q  approaches  the  point  P,  or  as  h  approaches  zero, 
X  does  not  change.     Then 

slope  of  e-^  at  P  =  e-  J^'J  [^"T^] "  ^2) 

We  now  seek  to  find 

limit  ["e'^  -  11 

h=Ol     h    J 
if  such  limit  exists. 

Since  the  fraction  (e*  —  l)/h  does  not  contain  x,  its  value,  for 
any  value  of  h,  and  hence  its  hmit,  is  the  same  for  every  point  of 
the  curve.  If  its  value  is  calculated  for  any  particular  point  of  the 
curve,  as  the  point  M,  it  will  have  this  value  at  any  other  point  as 
P.  .From  equation  (2)  the  slope  of  the  tangent  line  at  the  point 
M  is 

„  limitre^-1] 
^    ^=0L     h      J 
or 

limit  fe'-  -  1] 

h=Ol     h      i' 

But  by  the  definition  of  e,  the  slope  of  ?/  =  e*  at  M  is  1 . 

S[^]-'-  «> 

Substituting  this  result  in  equation  (2),  we  have 

slope  at  P  =  e'.  (4) 


262        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§146 


This  expresses  the  fact  that  the  slope  oi  y  =  e'  at  any  point  is  e", 
or  is  the  ordinate  y  of  that  point,  a  fact  that  was  first  indicated 
experimentally  in  §129. 
Later  an  approximate  value,  2.7183,  will  be  found  for  e. 

146.  Comparison  of  the  Curves  y.  =  v  and  y  =  e*.  In  Fig. 
105  the  slope  oiy  =  e'  atP  is  given  by  DP  measured  by  the  unit 
OM.  The  distance  TD,  the  subtangent,  is  constant  for  all  posi- 
tions of  the  point  P-    We  shall  prove  two  theorems. 

1.  The  curve  for  y  =  r"  can  be  made  from  y  =  e'  by  multiplying 
all  of  the  abscissae  of  the  latter  by  a  constant.    There  is  a  number  m 

such  that  e"*  =  r'.  Hence 
y  =  r'  may  be  written 
y  =  (e"')'  =  e""'.  Now 
this  curve  is  made  from 
2/  =  e*  by  substituting  mx 
for  X,  or  by  multiplying 
all  of  the  abscissas  of 
y  =  e'  hy  1/m. 

2.  The  slope  ofy  =  r'  at 
any  point  is  a  constant  times 
the  ordinate  of  that  point. 
The  curve  y  =  r"  can  be 
made  from  y  =  e'  hy  mul- 
tiplying all  of  the  abscissas 
of  the  latter  by  1/m. 
Therefore  the  side  TD  of 
the  triangle  PDT  va.  Fig. 
105  will  be  multiplied  by 
1/m,  the  other  side  DP 
remaining     the    same. 


T) 

o 

IS 

1 

17 

7\ 

1 

« 

a 

u 

1^ 

H 

s 

3 

14 

B 

IS 

B 

A 

n 

11 

in 

1 

7 

-n 

s 

/ 

1 

4 

/ 

\ 

3 

/ 

H 

=  log.  X    1 

- 

-^ 

— 

^ 

x=ey 

.^ 

-4 

-3 

-2 

-1 

-1 

/ 

!fs 

? 

3 

4 

5 

6 

7 

r 

9 

0 

1112  .3 

14 

-9, 

f 

^ 

-  2/.log,  *| 

-S 

— 

_ 

^ 

-4 

^' 

e' 

-el 

Fig.  106. — Exponential  and  logarith- 
mic curves  to  the  natural  base  e  =  log,,  x 
2.7183. 


Hence  the  slope  of  the  curve,  or  DP/TD,  will  be  multiplied  by 
m,  since  the  denominator  of  this  fraction  is  multiplied  by  1/m. 
Hence,  the  slope  oiy  =  r'  at  any  point  is  m  times  the  ordinate  of 
that  point,  where  m  satisfies  the  equation  e"  =  r. 

The  curve  y  =  e-'  is  (See  §25)  the  curve  y  =  e''  reflected  in 
the  y-axis.  This  curve,  as  well  as  the  curve  y  =  log,  x  and  its 
symmetrical  curve,  are  shown  in  Fig.  106.  Sometimes  the  curve 
y  =  e"  Ss.  called  the  exponential  curve  and  the  curve  y  =  log.  x 


§146]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  263 

is  called  the  logarithmic  curve.  This  distinction,  however,  has 
little  utility,  as  the  equation  of  either  locus  can  be  expressed  in 
either  notation. 

The  notation  y  =  In  x  is  often  used  to  indicate  the  natural  loga- 
rithm of  X  and  the  notation  y  =  Ig  x,  or  y  =  log  x,  is  used  to  stand 
for  the  common  logarithm  of  x. 

Table  IV. 

The  following  table  of  powers  of  e  is  useful  in  sketching  exponen- 
tial curves. 


eo.2  =  1.2214 

ei^  - 

=  1 

6487 

e-0.2 

=  0.8187 

e»-^  =  1.4918 

eM  = 

=  1 

3956 

g-0,4 

=  0.6703 

e»«  =  1.8221 

e!-4  = 

=  1 

2840 

g-0.6 

=  0.5488 

e»'  =  2,2256 

g-0.8 

=  0.4493 

e      =    2.7183 

e-' 

=  0.3679 

e^     =    7.3891 

e-'- 

=  0.1353 

e'  =  20.0855 

e-3 

=  0.0498 

e*     =   54.5982 

e-" 

=  0.0183 

\ 

/ 

\ 

\\ 

. 

/ 

/ 

?n=  0 

^ 

fc: 

m  =  0 

w 

% 

^fe 

-2-10  12 

Fig.  107. — A  family  of  exponentials,  y  =  e" 


Exercises 

1.  Draw  the  curve  y  =  e'-  +  e~^.     Show  that  y  is  an  even  function 
of  X,  that  is,  that  y  does  not  change  when  the  sign  of  x  is  changed. 

2.  Draw    the    curve    y  =  e"  —  e'".     Show  that  y  is  an  odd  fun  " 


264        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§147 

tion  of  X,  that  is,  that  the  function  changes  sign  but  not  absolute  value 
when  the  sign  of  x  is  changed. 

3.  Draw  the  graphs  oi  y  =  e",  and  y  =  e~''. 

4.  Draw  the  graphs  of  y  =  e*/*,  and  y  =  e~'/'. 

6.  Compare  the  curves:  y  =  e*/*,  y  =  e*''*,  y  =  e',  y  =  e'". 
6.  Sketch  the  curves  y  =  1',  y  =  2',  y  =  Z',  y  =  i",  y  =  5',  y  =  &', 
y  =  8',y  =  10"^,  from  a;=-3toa;=+3. 

147.  Change  of  Base  and  Properties  of  the  Exponential  Curv& 

Consider  the  curves  y  =  e'  and  y  =  a',  Fig.  108,  where  a  >  e. 
For  a  given  y  =  OH,  the  abscissas  HP\  and  ffiPz  are  log,  y  and 
logaj/,  respectively.  It  has  been  shown  (§146)  that  the  curve 
y  =  a'  can  be  obtained  from  the  curve  y  =  e*  by  multiplying  the 

abscissas  of  the  latter  curve  by  — ,  where  m  is  the  number  such  that 

■  a.  (1) 


T 

"/ 

* , 

r 

»/ 

"/ 

4/ 

y 

B 

* 

2=^ 

Jt> 

.yp 

^ 

/> 

7^P\ 

0 

In  other  words. 
That  is, 


EPi  =  -  EP^. 

m 


log.  y  =  -  log'  y- 


(2) 


■•^  As   soon  as  m  is  known  we  have  a 
means  of  changing  from  a  systeni  of 
^"*-  I08--Co°iparison  of  logarithms  with  base  e  to  one  with  base 

V  =  er  and  y  =  a".  °  . 

a.     The  number  —  is  called  the  modulus 
m 

of  the  logarithmic  system  whose  base  is  a. 

The  modulus  of  the  common  system  of  logarithms  is  represented 

by  M.    It  is  the  value  of  —  where  m  satisfies 


e">  =  10,  or  TO  =  log,  10, 


(3) 


which  is  equation  (1)  for  a  =  10. 
That  is. 


Hence, 


e'^  =  10,  or  e  =  10". 
M  =  logio  e  =  0.4343. 


(4)' 


§147]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  265 

Hence,  if  N  represents  any  number, 

logio  N  =  0.4343  log.  N,  (5) 

log.N  =  — - — logioN  =  2.3026  log,  N.  (6) 

04343 

From  the  definition  of  m  and  M, 

100.«43  =  g^  (7) 

g2.3026  =  10.  (8) 

Incidentally  it  should  be  noted   that,   since   M  =  —'  from  (3) 
and  (4), 

^°S^°"  =  1^'  ^^^ 

A  remarkable  property  of  the  logarithmic  curve  appears  from 
comparing  the  curves  y  =  a'  and  y  =  a''*'''.  The  second  of  these 
curves  can  be  derived  from  y  =  a' by  translating  the  latter  curve 
the  distance  h  to  the  left.  But  y  =  a*+''  may  be  written  y  =  a'^a^, 
from  which  it  can  be  seen  that  the  new  curve  may  also  be 
considered  as  derived  from  y  =  a'  hy  multipljang  aU  ordinates 
oiy  =  a"  by  a*. 

Translating  the  exponential  curve  in  the  negative  x-direction  is  the 
same  as  multiplying  all  ordinaies  by  a  certain  fixed  number,  or  is 
equivalent  to  a  certain  orthographic  projection  of  the  original  curve 
upon  a  plane  through  the  X-axis. 

Changing  the  sign  of  h  changes  the  sense  of  the  translation  and 
changes  elongation  to  shortening  or  vice  versa. 

Exercises 

1.  Compare  the  curve  y  =  e"  with  the  curve  y  =  10*. 

2.  Graph  the  logarithmic  spiral  p  =  e>,6  being  measured  in  radians. 
Note  :  The  radian  measure  in  the  margin  of  Form  MZ  should  be 

used  for  this  purpose. 

3.  Graph  p  =  e-«. 

4.  The  pressure  of  the  atmosphere  is  given  in  millimeters  of  mer- 
cury by  the  formula 

y  =  760.e-»'/'i""' 
where  the  altitude  x  is  measured  in  meters  above  the  sea  level.     Pro- 


266        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§148 


duce  a  table  of  pressure  for  the  altitudes  x  =0;  10;  50;  100;  200; 
300;  1000;  10,000;  100,000. 

5.  From  the  data  of  the  last  problem,  find  the  approximate  pressure 
at  an  altitude  of  25,000  feet. 

6.  Show  that  the  relation  of  exercise  4  may  be 
written 

X  =  18,421  (log  760  -  log  y). 

7.  Determine  the  value  of  the  quotient  j for 

the  following  values  of  x :  2,  3,  5,  7. 

8.  How  large  is  e"""',  approximately? 

9.  What  is  the  approximate  value  of  lO"""!? 

148.  Logarithmic  Double  Scale.  The  relation 
between  a  number  and  its  logarithm  can  be 
shown  by  a  double  scale  of  the  sort  discussed  in 
§§3  and  10.  Such  a  scale  is  shown  in  Fig.  109. 
It  may  be  constructed  as  follows:  First  con- 
struct the  uniform  scale  A,  in  which  the  unit 
distance  0  —  1  is  shown  divided  into  100  equal 
parts.  Opposite  0.3010  (  =  log  2)  of  the  A- 
scale  place  a  division  line  on  the  5-scale  marked 
by  the  number  2.  Opposite  0.4771  (=  log  3) 
of  the  A-scale  place  a  division  line  of  the  B- 
scale  marked  by  the  number  3.  Likewise  op- 
posite 0.6021  (=  log  4)  of  A  place  4  on  B;  op- 
posite 0.6990  (=  log  5)  of  A  place  5  on  B;  etc. 
Intermediate  points  on  B  are  similarly  located — 
for  example  the  2.1  mark  on  B  should  be 
placed  opposite  0.3222  (=  log  2.1)  on  A. 

The  non-uniform  scale  B  is  called  a  loga- 
rithmic scale,  for  the  lengths  measured  along  it 
are  proportional  to  the  logarithms  of  the  natural 
numbers. 

The  double  scale  of  Fig.  109  may  obviously 
be  used  as  a  table  of  logarithms.    Thus  from 

it  we  may  read  log  7.1  =  0.85;  log  3.3  =  0.52;  log  1.5  =  0.175. 
Since  log  10a;  =  1  +  log  x,  it  follows  that,  if  the  scales  A  and  B, 

Fig.  109,  were  extended  another  unit  to  the  right,  this  second 


s      — 


3 


a 
.a 


a 
ho 
o 


=      i! 


§149]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  267 

unit  would  be  identical  to  the  first  one,  except  in  the  attached 
numbers.  The  numbers  on  the  A-scale  would  be  changed  from 
0.0,  0.1,  0.2,  .  .,1.0  to  1.0,  1.1,  1.2,  .  ,,  2.0,  while  those 
on  the  non-uniform,  or  S-scale,  would  be  changed  from  1,  2,  3, 
.    .    .,  10  to  10,  20,30,      .    .,  100. 

Passing  along  this  scale  an  integral  number  of  unit  intervals 
corresponds  thus  to  change  of  characteristic  in  the  logarithms,  and 
to  change  in  the  position  of  the  decimal  point  in  the  numbers.   ■ 

It  is  not,  however,  necessary  to  construct  more  than  one  block  of 
this  double  scale,  since  we  are  at  liberty  to  add  an  integer  n  to  the  num- 
bers of  the  uniform  scale,  provided  at  the  same  time  we  multiply  the 
numbers  of  the  non-uniform  scale  by  10".  In  this  way  we  may  obtain 
any  desired  portion  of  the  extended  scale.  Thus,  we  may  change  0.1, 
0.2,  0.3,  .  .,  1.0  on  X  to  3.1,  3.2,  3.3,  .  .  .,  4.0,  by  adding  3  to 
to  each  number,  provided  at  the  same  time  we  change  the  numbers 
1,  2,  3,  4,  .  .,  10  on  the  S-scale  to  1000,  2000,  3000,  4000,  .  ., 
10,000  by  multiplying  them  by  10^.  If  n  be  negative  (say  —  2)  we 
may  write,  as  in  the  case  of  logarithms,  8.0  —  10,  8.1  —  10,  8.2  — 
10,  .  .  .,  9.0  -  10,  or,  more  simply,  -  2,  -  1.9,  -  1.8,  -  1.7, 
.,  —  1.0,  changing  the  numbers  on  the  non-uniform  scale  at  the 
same  time  to  0.01,  0.02,  0.03,       .    .,  0.10. 

Exercises 

Read  the  following  from  the  double  scale.  Fig.  109. 

1.  log  5.5  2.  log  2.4  3.  log  1.9  4.  log  71 

6.  anti-log'O.74     6.  anti-log  0.38   7.  anti-log  1.38   8.  anti-log  2.38 

149.  The  Slide  Rule.  By  far  the  most  important  apphcation 
of  the  non-uniform  scale  ruled  proportionally  to  log  z,  is  the  com- 
puting device  known  as  the  slide  rule.  The  principle  upon  which 
the  operation  of  the  slide  rule  is  based  is  very  simple.  If  we  have 
two  scales'  divided  proportionally  to  log  x  (A  and  B,  Fig.  110), 
so  arranged  that  one  scale  may  slide  along  the  other,  then  slid- 
ing one  scale  (called  the  slide)  until  its  left  end  is  opposite  any 
desired  division  of  the  first  scale,  selecting  any  desited  division  of 
the  slide,  as  at  R,  Fig.  110,  and  taking  the  reading  of  the  original 
scale  beneath  this  point,  as  N,  the  product  of, the  two  factors 
whose  logarithms  are  proportional  to  AB  and  BR  can  be  read 


268        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§149 


X 


directly  from  the  lower  scale  at  N.  For  AN  is,  by  construction, 
the  sum  oi  AB  and  BR,  and  since  the  scales 
were  laid  off  proportionally  to  log  x  and  marked 
with  the  numbers  of  which  the  distances  are  the 
logarithms,  the  process  described  adds  the  loga- 
rithms mechanically,  but  indicates  the  results 
in  terms  of  the  numbers  themselves.  By  this 
device  all  of  the  operations  commonly  carried 
out  by  use  of  a  logarithmic  table  may  be  per- 
formed mechanically.  Full  description  of  the 
use  of  the  shde  rule  need  not  be  given  in  de- 
tail at  this  place,  as  complete  instructions  are 
found  in  the  pamphlet  furnished  with  each 
slide  rule.  A  very  brief  amount  of  individual 
instruction  given  to  the  student  by  the  instruc- 
tor will  insure  the  rapid  acquirement  of  skill  in 
the  use  of  the  instrument.  In  what  follows, 
the  four  scales  of  the  slide  rule  are  designated 
from  top  to  bottom  of  the  rule,  hy  A,  B,  C,  D, 
respectively.  The  ends  of  the  scales  are  called 
the  indices. 


AH|Ordinary  10-inch  sUde  rule  should  give  results 
accurate  to  three  significant  figures,  which  is  ac- 
curate enough  for  most  of  the  purposes  of  applied 
science. 

An  exaggerated  idea  sometimes  prevails  con- 
cerning the  degree  of  accuracy  required  by  work 
in  science  or  in  applied  science.  Many  of  the 
fundamental  constants  of  science,  upon  which  a 
large  number  of  other  results  depend,  are  known 
only  to  three  decimal  places.  In  such  cases 
greater  than  three  figure  accuracy  is  impossible 
even  if  desired.  In  other  cases  greater  accuracy 
is  of  no  value  even  if  possible.  The  real  deside- 
ratum in  computed  results  is,  first,  to  know  by  a 
suitable  check  thai  the  work  of  compiUation  is  correct, 
and,  second,  to  know  to  what  order  or  degree  of  ac- 
curacy both  the  daia  and  the  resuU  are  dependahle. 

The  absurdity  of  an  undue  number  of  decimal 


^-2!*  » 


j3 


a 


§149]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  269 

places  in  computation  is  illustrated  by  the  original  tables  of  loga- 
rithms, which  if  now  used  would  enable  one  to  compute  from  the 
radius  of  the  earth,  the  circumference  to  1/10,000  ■part  o/  an  inch. 

The  following  matters  should  be  emphasized  in  the  use  of  the 
slide  rule : 

(1)  All  numbers  for  the  purpose  of  slide  rule  computation  should 
be  considered  as  given  with  the  first  figure  in  units  place.  Thus 
517  X  1910  X  0.024  should  be  considered  as  5.17  X  1.19  X  2.4  X 
10^  X  10'  X  10~^  The  result  should  then  be  mentally  approxi- 
mated (say  24,000)  for  the  purpose  of  locating  the  decimal  point, 
and  for  checking  the  work. 

(2)  A  proportion  should  always  be  solved  by  one  setting  of  the 
slide. 

(3)  A  combined  product  and  quotient  like 

aXhXcXd 
■     rXsXt 

should  always  be  solved  as  follows: 

Place  runner  on  a  of  scale  D; 

set  r  of  scale  C  to  a  of  scale  D; 

runner  to  fe  of  C; 

s  of  C  to  rurmer; 

runner  to  c  of  C; 

t  of  C  to  runner; 

runner  to  d  of  C;  find  on  D  the  significant  figures  of  the  ' 

result. 

(4)  The  runner  must  be  set  on  the  first  half  of  A  for  square 
roots  of  numbers  having  an  odd  number  of  digits,  and  on  the 
second  half  of  A  for  the  square  roots  of  other  numbers. 

(5)  Use  judgment  so  as  to  compute  results  in  most  accurate 
manner — thus  instead  of  computing  264/233,  compute  31/233  and 
hence  find  264/233  =  1  +  31/233.^ 

(6)  Besides  checking  by  mental  calculation  as  suggested  in  (1) 
above,  also  check  by  computing  several  neighboring  values  and 
graphing  the  results  if  necessary.  Thus  check  5.17  X  1.91  X  2.4 
by  computing  both  5.20  X  19.2  X  2.42  and  5.10  X  1.90  X  2.38. 

1  Show  by  trial  that  this*  gives  a  more  accurate  result. 


270        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§149 


Exercises 

Compute  the  following  on  the  slide  rule: 

1.  3.12  X  2.24;    1.89  X  4.25;    2.88  X  3.16;    3.1  X  236. 

2.  8.72/2.36;    4.58/2.36;    6.23/2.12;    10/3.14. 

3.  32.5  X  72.5;    0.000116  X  0.00135;    0.0392/0.00114. 

4.  3,967,000  H-  367,800,000.  g    78.5  X  36.6  X  20.8 
,    6.64X42.6   8.75X5.25  '  „  .^■'^^J^?^^ 


32.5  '       32.3 

Solve  the  proportion 


6.46  X  57.5  X  8.55 
3.26  X296  X  0.642' 


X  :  1.72  =  4.14  :  V^gh. 
where  g  =  32.2  andA^  =  78.2. 
o    n  ,    VlTl  X  1.41 

9.  Compute    166.7X4.5' 

10.  The  following  is  an  approximate  formula  for  the  area  of  a  seg- 
ment of  a  circle :  <• 

A  =  h'/2c  +  2ch/3, 

where  c  is  the  length  of  the  chord  and  h  is  the  altitude  of  the  segment. 
Test  this  formula  for  segments  of  a  circle  of  unit  radius,  whose  arcs 
are  7r/3,  ir/2,  and  tt  radians,  respectively. 

11.  Two  steamers  start  at  the  same  time  from  the  same  port;  the 
first  sails  at  12  miles  an  hour  due  south,  and  the  second  sails  at  16 
miles  an  hour  due  east.  Knd  the  bearing  of  the  &st  steamer  as  seen 
from  the  second  {l)  after  one  hour,  (2)  after  two  hours,  and  compute 
their  distances  apart  at  each  time. 

The  following  exercises  require  the  use  of  the  data  printed  herewith. 
An  "acre-foot"  means  the  quantity  of  water  that  would  cover  1 
acre  1  foot  deep.  "Second-foot"  means  a  discharge  at  the  rate  of  1 
cubic  foot  of  water  per  second.  By  the  "run-off"  of  any  drainage 
area  is  meant  the  quantity  of  water  flowing  therefrom  in  its  surface 
stream  or  river,  during  a  year  or  other  interval  of  time. 

1  square  mUe  =  640  acres. 

1  acre  =  43,560  square  feet. 

1  day  =  86,400  seconds. 

1  second-foot  =  2  acre-feet  per  day,  approximately. 

1  cubic  foot  =  7J  gallons,  approximately. 

1  cubic  foot  water  =  62^  pounds  water,  approximately. 

1  h.p.  =  550  foot-pounds  per  second. 

450  gallons  per  minute  =  1  second-foot,  approximately. 


§150]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  271 

Each  of  the  following  problems  should  be  handled  on  the  slide  rule  as 
a  continuous  piece  of  computation. 

12.  A  drainage  area  of  710  square  miles  has  an  annual  run-off  of 
120,000  acre-feet.  The  average  annual  rainfall  is  27  inches.  Find 
what  percent  of  the  rainfall  appears  as  run-off. 

13.  A  centrifugal  pump  discharges  750  gallons  per  minute  against 
a  total  lift  of  28  feet.  Find  the  theoretical  horse  power  required. 
Also  daily  discharge  in  acre-feet  if  the  pump  operates  fourteen  hours 
per  day. 

14.  What  is  the  theoretical  horse  power  represented  by  a  stream 
discharging  550  second-feet  if  there  be  a  fall  of  42  feet? 

15.  A  district  containing  25,000  acres  of  irrigable  land  is  to  be  sup- 
plied with  water  by  means  of  a  canal.  The  average  annual  quantity 
of  water  required  is  Sf  feet  on  each  acre.  Find  the  capacity  of  the 
canal  in  second-feet,  if  the  quantity  of  water  required  is  to  be  delivered 
uniformly  during  an  irrigation  season  of  five  months. 

16.  A  municipal  water  supply  amounts  to  35,000,000  gallons  per 
twenty-four  hours.     Find  the  equivalent  in  cubic  feet  per  second. 

17.  A  single  rainfall  of  3.9  inches  on  a  catchment  area  of  210  square 
mUes  is  found  to  contribute  17,500  acre-feet  of  water  to  storage  reser- 
voir.    The  run-off  is  what  percent  of  the  rainfall  in  this  case? 

150.  Semi-logarithmic  Coordiaate  Paper.  Fig.  Ill  represents 
a  sheet  of  rectangular  coordinate  paper,  on  which  ON  has  been 
chosen  as  the  unit  of  measure.  Along  the  right-hand  edge  of  this 
sheet  is  constructed  a  logarithmic  scale  LM  of  the  type  discussed 
in  §148,  i.e.,  any  number,  say  4,  on  the  scale  LM  stands  opposite 
the  logarithm  of  that  number  (in  the  case  named  opposite  0.6021) 
on  the  uniform  scale  ON. 

Let  us  agree  always  to  designate  by  capital  letters  distances 
measured  on  the  uniform  scales,  and  by  lower  case  letters  dis- 
tances measured  on  the  logarithmic  scale.  Thus  Y  will  mean  the 
ordinate  of  a  point  as  read  on  the  scale  ON,  while  y  will  mean  the 
ordinate  of  a  point  as  read  on  the  scale  LM.  Moreover,  we  agree 
to  plot  a  function,  using  logarithms  of  the  values  of  the  function 
as  ordinates  and  the  natural  values  of  the  argument,  or  variable, 
as  abscissas. 

Let  PQ  be  any  straight  line  on  this  paper,  and  let  it  be  required 
to  find  its  equation,  referred  to  the  uniform  a;-scale  OL  and  the 
logarithmic  2/-scale  LM.    We  proceed  as  follows : 


272        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§150 


The  equation  of  this  line,  referred  to  the  uniform  Z-axis  OL 
and  the  uniform  Y-axis,  ON,  where  0  is  the  origin,  is 

Y  =  mx  +  B, 

m  being  the  slope  of  the  line,  and  B  its  F-intercept.    ,Now,  for  the 

line  PQ,  m  =  0.742  and  B  =  0.36,  so  that  the  equation  of  PQ  is 

Y  =  0.742a:  +  0.36.  (1) 

To  find  the  equation  of  this  curve  referred  to  the  scales  LM  and 

OL,  it  is  only  necessary  to  notice  that 

y  =  log  2/ 


;v 

Q 

I 

% 

1.U 

/^ 

/ 

9 

.8 
.7 
.6 

y 

7 

/ 

y 

6 
S 

Y 

/ 

y 

y 

3 

< 

i2 

IL 

.1  .2  .3         .4  .5  .6  .7         .8  .9         1.0^ 

Fig.  111.— The  theory  of  the  use  of  semi-logarithmic  paper. 

so  that  we  obtain 

log  y  =  0.742a;  +  0.36.  (2) 

The  intercept  0.36  was  read  on  the  scale  ON,  and  is  therefore  the 
logarithm  of  the  number  corresponding  to  it  on  the  scale  LM. 
That  is,  0.36  =  log  2.30.  Substituting  this  value  in  equation 
(2)  we  obtain 

log  V  =  0.742a;  +  log  2.30, 


§150]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  273 

which  may  be  written 

log  y  -  log  2.30  =  0.742a;, 


log  2|o  =  0-7*2.. 


}^M     *='         ^ 

cS            c 

=■        °     o"'  w-° 

c 

/ 

1, 

/ 

. 

/ 

8 

/ 

y 

7 

y 

0 

y 

' 

6  = 

/ 

b 

y 

/^ 

4 

ny 

/ 

S 

/ 

•^ 

S 

/ 

y 

4 

2 

2 

1 

0.1 

n 

A    L    0.1       0.2       0:3       0.4       0.5       o.<       0.7       0.8       0.0      \,0B 
Semi  LosarTthmlc  Paper 

Fig.  112, — Illustration  of  squared  paper,  form  M5.     The  finer  rulings 
of  form  M5  have,  however,  been  omitted. 

Changing  to  exponential  notation  this  becomes 


2.30 


10»' 


y  =  2.30(10°'«»).  (3) 

In  general,  if  the  equation  of  a  straight  line  referred  to  the  scales 
OL  and  ON  is 

F  =  m  +  S,      ,  (4) 

18 


274        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§150 

its  equation  referred  to  the  scales  OL  and  LM  may  be  obtained  by 
replacing  Y  by  log  y  and  B  by  log  6  in  the  manner  described  above, 
giving 

log  y  =  mx  +  log  6,  (5) 

which,  as  above,  may  be  reduced  to  the  form 

y  =  bio"*.  (6) 

This  is  the  general  equation  of  the  exponential  curve.  Hence: 
Any  exponential  curve  can  he  represented  by  a  straight  line,  provided 
ordinates  are  read  from  a  suitable  logarithmic  scale,  and  abscissas 
are  read  from  a  uniform  scale. 

Fig.  112  represents  the  same  line  PQ,  y  =  (2.30)10°'"',  as 
Fig.  111.  The  two  figures  differ  only  in  one  respect;  in  Fig.  Ill 
the  rulings  of  the  uniform  scale  ON  are  extended  across  the  page, 
while  in  Fig.  112  these  rulings  are  replaced  by  those  of  the  scale 
LM. 

Coordinate  paper  such  as  that  represented  by  Fig.  112  is  known 
as  semi-logarithmic  paper.  It  affords  a  convenient  coordinate 
system  for  work  with  the  exponential  function. 

Every  point  on  PQ  (Fig.  112)  satisfies  the  exponential  equation 

y  =  2.30(10"  '^2-). 
Thus,  in  the  case  of  the  point  R, 

3.98  =  2.30(10'''")'''2» 
=  2.30(10»-238). 

The  slope  of  any  line  on  the  semi-logarithmic  paper  may  be  read 
or  determined  by  means  of  the  uniform  scales  BC  and  AB  oi  form 
M5.  The  scale  AD  of  form  M5  is  the  scale  of  the  natural  loga- 
rithms, so  that  any  equation  of  the  form  y  =  e""  can  be  graphed 
at  once  by  the  use  of  this  scale.  Thus,  the  line  y  =  e"''(Fig. 
113)  passes  through  the  point  A  or  (0,  1),  and  a  point  on  BC  op- 
posite the  point  marked  1.0  on  AD.  Note  that  1.0  on  scale  AD, 
-  2.718  on  the  non-uniform  scale  of  the  main  body  of  the  paper, 
and  0.4343  on  the  scale  BC  aU  fall  together,  as  they  should. 

To  draw  the  line  y  =  10""',  the  corner  D  of  the  plate  may  be 
taken  as  the  point  (0, 1).  On  the  line  drawn  once  across  the  sheet 
representing  y  =  10"*,  y  has  a  range  between  1  and  10  only.* 


§150]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  275 

To  represent  the  range  of  y  between  10  and  100,  two  or  more  sheets 
of  form  M5  may  be  pasted  together,  or,  preferably,  the  continua- 
tion of  the  line  may  be  shown  on  the  same  sheet  by  suitably  chang- 
ing the  numbers  attached  to  the  scales  AB  and  BC.  Thus  Fig. 
113  shows  in  this  manner  y  =  IW". 


Il  1 1 1 1 1 1 1 1 



1, 

,,,, 



9\ 

/,* 

\ 

/ 

/  ! 

8       \ 

/    I    R 

\ 

/ 

7 

N, 

/ 

/ 

/      ' 

S, 

/ 

. 

6 

V 

/ 

/ 

g 

\, 

1 

/ 

/ 

5 

\ 

1 

/ 

/ 

6 

V 

/ 

/ 

1 

4\ 

/ 

1 

4 

/ 

/ 

a 

\ 

/ 

/ 

8 

^ 

/ 

3 

1 
1 
t 

/ 

\ 

/ 

^ 

a 

/ 

A 

>> 

■i 

^ 

/ 

/ 

/ 
I 
I 
t 

/ 

/ 

^ 

< 

\ 

s 

1 

^ 

/ 

^ 

'^ 

1 
1 
1 

1 

1 

1 
1 
1 

\ 

\ 

\ 

.1  0.1         0.2        0.3        0.4         0.5        0.6        0.7         0.8        0.9       l.OB 

Fig.  113. — Seini-logarithmic  coordinate  paper.    The  dotted  line 
gives  two  sections  oi  y=  10^"^. 

Remember  that  on  semi-logarithmic  paper  the  line 

y  =  bio""^  (7) 

passes  through  the  point  (0,  6)  with  slope  m.     Note  that 


f  =  lo^C*  -  ») 


(8) 


passes  through  the  point  (a,  6)  with  slope  m. 

Illtjstration  1.  Draw  the  curve  ^x  =  log  ^y  on  semi-logarithmic 
paper. 

This  is  the  curve  y  =  2(105"^).  This  curve  passes  through  the  point 
(0,  3)  with  slope  |,  hence  can  readily  be  drawn. 


276        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§150 _ 

lUiTTBTBATiON  2.     Draw  the  curve  y  =  3(10'^""*^]. 

From  (8)  above  it  is  seen  that  this  curve  passes  through  the  point 
(2,  3)  with  slope  2. 

Illustration  3.  Plot  the  following  data  upon  semi-logarithmic 
paper  and  find,  if  possible,  the  equation  connecting  the  x-  and  y- 
values. 


10 

8 

I 

^ 

^ 

4 
S 

2 

1 

^ 

• 

> 

^ 

^ 

^ 

^ 

0.2         0.3 

Fig.  114.- 


0  4      0  5     0.6      0.7     0.8      as 
-Diagram  for  Illustration  3. 


X 

y 

0.2 

3.18 

0.4 

3.96 

0.6 

5.00 

0.8 

6.30 

The  points  plotted  upon  semi-logarithmic  paper  he  on  a  straight 
line  as  shown  in  Fig.  114.    Hence,  it  is  possible  to  find  the  equation 
connecting  x  and  y.     The  equation  of  this  straight  line  is 
Y  =  \x^r  log  2.51, 


§151]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  277 


or 
or 


log  V  =  hx  -\-  log  2.51, 

'•  y   _  ^ 


log 


2.51       2' 


y 

2.51 


=  10' 


V  =  2.51(10»), 
an  empirical  equation  connecting  the  x-  and  j^-values  of  the  table. 


Exercises 

On  semi-logarithmic  paper  draw  the  following: 

1.  y  =  10",  y  =  W,  y  =  10»^. 

2.  y  =  10-==,  y  =  IQ-'i*,  y  =  10-»«. 

3.  y  =  e^',  y  =  e". 
i.  y  =  e"",  y  =  e~^'. 

5.  Sx  =  log  y,  (l/2)a;  =  log  y. 

6.  Find  an  empirical  equation  connecting  the  x-  and  the  y-values 
given  in  the  accompanying  table. 


X 

y 

0.2 
0.4   , 
0.6 
0.8 

5.8 

3.4 
2.6 

On  semi-logarithmic  paper  draw  the  following: 
^.  y  =  10'/2,  y  =  io»/io. 

■  8.  Graph  y  ^  2(10)'  and  |  =  10'-'. 

161.*  The  Compound  Interest  Law.    Computation  of  e.    The 

law  expressed  by  the  exponential  curve  was  called  by  Lord  Kelvin 
the  compound  interest  law  and  since  that  time  this  name  has 
been  .generally  used.  It  is  recalled  that  the  exponential  curve 
was  drawn  by  using  ordinates  equal  to  the  successive  terms  of 
a  geometrical  progression  which  are  uniformly  spaced  along  the 


278        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§151 

X-axis.  Since  the  amount  of  any  sum  at  compound  interest  is 
given  by  a  term  of  a  geometrical  progression,  it  is  obvious  that  a 
sum  at  compound  interest  accumulates  by  the  same  law  of  growth 
as  is  indicated  by  a  set  of  uniformly  spaced  ordinates  of  an  expo- 
nential curve;  hence  the  term  "compound  interest  law,"  from 
this  superficial  view,  is  appropriate.  The  detailed  discussion 
that  follows  will  make  this  clear: 

Let  $1  be  loaned  at  r  percent  per  annum  compound  interest. 
At  the  end  of  one  year  the  amount  is  (1  -|-  r/100). 
At  the  end  of  two  years  the  amount  is  (1  -|-  r/100)'', 
■  and  at  the  end  of  t  years  it  is  (1  +  r/100)'. 
If  interest  be  compounded  semi-annually,  instead  of  annually, 
the  amount  at  the  end  of  t  years  is  (1  -|-  r/200)''', 
and  if  compounded  monthly  the  amount  at  the  end  of  the  same 
period  is  (1  +-r/\2QQy^' 

or  if  compounded  n  times  per  year  y=  {1  +  r/lOOn)"', 
where  t  is  expressed  in  years.  Now  if  we  find  the  limit  of  this 
expression  as  n  is  increased  indefinitely,  we  will  find  the  amount  of 
principle  and  interest  on  the  hypothesis  that  the  interest  was 
compounded  conlinuously .  For  convenience  let  r/lOOn  =  1/m. 
Then 

2/  =  (1  +  1/m)-'/'»»,  (1) 

where  the  limit  is  to  be  taken  as  m  or  n  becomes  infinite.     Calling 

(1  +  l/uY  =  f(u)  (2) 

and  expanding  by  the  binomial  theorem  for  any  integral  value 

of  u  we  obtain 

r/-  \       II      fi  /  \   i  w(w  —  1)    1    , 

f{u)  =  1  +  u{l/u)  +     \2        ^  +   •    ■    • 

=  1  -I-  1  +  (1  -  l/w)/2!  -I-  (1  -  1/m)(1  -  2/u)/3\  -I-  ...    (3) 

In  the  calculus  it  is  shown  that  the  limit  of  this  series  as  u  becomes 
infinite  is  the  limit  of  the  series 

l-hl  +  l/2!-M/3!-h  ...  (4) 

The  limit  of  this  series  is  easily  found;  it  is,  in  fact,  the  Naperian 
base  e.  It  is  shown  in  the  calculus  that  the  restriction  that  u 
shall  be  an  integer  may  be  removed,  so  that  the  limit  of  (3)  may 
be  found  when  m  is  a  continuous  variable. 


§152]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  279 

It  is  easy  to  see  that  the  Umit  of  (4)  is  >  2^  and  <  3.  The  sum 
of  the  first  three  terms  of  the  series  (4)  equals  2i;  the  rest  of  the 
terms  are  positive,  therefore  e>2^.  The  terms  of  the  series  (4), 
after  the  first  three,  are  also  observed  to  be  less,  term  for  term,  than 
the  terms  of  the  progression: 

(1/2)2  +  (1/2)3  +  (5) 

But  this  is  a  geometrical  progression  the  limit  of  whose  sum  is  1/2. 
Therefore  (3)  is  always  less  than  2|  +  5,  or  3.  The  value  of  e  is 
readily  approximated  by  the  following  computation  of  the  first 
8  terms  of  (4): 

2.00000  =  1  +  1 


0.50000  =  1/2! 
0.16667  =  1/3! 


0.04167  =  1/4! 
0.00833  =  1/5! 
0.00139  =  1/6! 


0.00020  =  1/7! 
Sum  of  8  terms  =  2.71826 

The  value  of  e  here  found  is  correct  to  four  decimal  places. 

Returning  to  equation  (1)  above,  the  amount  of  $1  at  r  percent 
compound  interest  compounded  continuously  is 

y  =  e"/""-  (6) 

Thus  $100  at  6  percent  compound  interest,  compounded  annually, 
amounts,  at  the  end  of  ten  years,  to 

y  =  100(1.06)"'  =  $179.10. 

The  amount  of  $100  compounded  continuously  for  ten  years  is 

y  =  100e«-6=  $182.20 

The  difference  is  thus  $3.10 

152.  Logarithmic  Increment.  The  compound  interest  law  is 
one  of  the  important  laws  of  nature.  As  previously  noted,  the 
slope  or  rate  of  increase  of  the  exponential  function 

y  =  ae'" 

at  any  point  is  always  proportional  to  the  ordinate  or  to  the  value 


280        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§152 

of  the  function  at  that  point.  Thiis  when  in  nature  we  find  any 
function  or  magnitude  that  increases  at  a  rate  proportional  to  itself 
we  have  a  case  of  the  exponential  or  compound  interest  law. 
The  law  is  also  frequently  expressed  by  saying,  as  has  been  re- 
peatedly stated  in  this  book,  that  the  first  of  two  magnitudes  varies 
in  geometrical  progression  while  a  second  magnitude  varies  in  arith- 
metical progression.  A  famUiar  example  of  this  is  the  increased 
friction  as  a  rope  is  coiled  around  a  post.  A  few  turns  of  the 
hawsers  about  the  bitts  at  the  wharf  is  sufficient  to  hold  a  large 
ship,  because  as  the  number  of  turns  increases 'In  arithmetical 
progression,  the  friction  increases  in  geometrical  progression. 
Thus  the  following  table  gives  the  results  of  experiments  to  de- 
termine what  weight  could  be  held  up  by  a  one-pound  weight, 
when  a  cord  attached  to  the  first  weight  passed  over  a  round  peg 
the  number  of  times  shown  in  the  first  column  of  the  table: 


Average  logarithmic  increment  = 


n  =  number  of 

turns  of  the  cord 

on  the  peg 

w  =  weight  juBt  held 
in  equilibrium  by 
one-pound  weight 

Logs  of  preceding 
numbers 

d  =  logarithmic 
increment 

1 

1.6 
3.0 
5.1 
8.0 
14.0 
23.0 

0.204 
0.477 
0.708 
0.903 
1.146 
1.362 

1 

li 

2 

21 

3 

0.273 
0.231 
0.195 
0.243 
0.216 

0.23 


If  the  weights  sustained  were  exactly  in  geometrical  progression, 
their  logarithms  would  be  in  arithmetical  progression.  The  test 
for  this  fact  is  to  note  whether  the  differences  between  logarithms 
of  successive  values  are  constant.  These  differences  are  known 
as  logarithmic  increments  or  in  .case  they  are  negative,  as  loga- 
rithmic decrements.  In  the  table  the  logarithmic  increments 
fluctuate  about  the  mean  value  0.23. 
The  equation  connecting  n  and  w  is  of  the  form 


w  =  10'"'"'  or  n  =  m  log  w 


§153]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  281 

By  graphing  columns  1  and  3  on  squared  paper,  the  value  of  m  is 
determined  and  we  find 

w  =  lO"-^'",  or  n  =  2.2  log  w. 

Another  way  is  to  graph  columns  1  and  2  on  semi-logarithmic 
paper. 

An  interesting  example  of  the  compound  interest  law  is  Weber's 
law  in  psychology,  which  states  that  if  stimuli  are  in  geometrical 
progression,  the  sense  perceptions  are  in  arithmetical  progression. 

163.  Modulus  of  Decay,  Logarithmic  Decrement.  In  a  very 
large  number  of  cases  in  nature  a  function  obeying  the  "compound 
interest"  law  appears  as  a  decreasing  function  rather  than  as  an 
increasing  function,  so  that  the  equation  is  of  the  form 

y  =  ae~>"',  (1) 

where  ( —  6)  is  essentially  negative,  b  is  the  modulus  of  decay,  or 
the  logarithmic  decrement,  corresponding  to  an  increase  of  x 
by  unity.    The  following  are  examples  of  this  law: 

(1)  If  the  thickness  of  panes  of  glass  increase  in  arithmetical  pro- 
gression, the  amount  of  light  transmitted  decreases  in  geometrical 
progression.     That  is,  the  relation  is  of  the  form 

L  =  oe-«,  (2) 

where  t  is  the  thickness  of  the  glass  or  other  absorbing  material  and  L 
is  the  intensity  of  the  light  transmitted.  Since  when  t  =  0  the  light 
transmitted  must  have  its  initial  intensity,  Lo,  (2)  becomes 

L  =  Loe-«.  (3) 

The  constant  6  must  be  determined  from  the  data  of  the  problem. 
Thus,  if  a  pane  of  glass  one  unit  thick  absorbs  2  percent  of  the  incident 
light, 

U  =  100,  Z,  =  98  for  «  =  1, 

and  98  =  100e-», 

or  log  98  -  log  100  =  -  6  log  e. 

Therefore  6  =  j^j^  =  0.02 

The  light  transmitted  by  ten  panes  of  glass  is  then 
iio  =  100e-"'f»»«)  =  100e-»-2, 


282        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§153 

or,  by  the  table  of  §146, 

Lio  =  100/1.2214  =  82  percent 

(2)  Variation  in  atmospheric  pressure  with  the  altitude  is  usually 
expressed  by  Halley's  Law, 

p  =  760e-*/8»»», 

where  h  is  the  altitude  in  meters  above  sea  level  and  p  is  the  atmos- 
pheric pressure  in  millimeters  of  mercury.     See  §147,  Exercises  4,  5,  6. 

(3)  Newton's  law  of  cooling  states  that  a  body  surrounded  by  a 
medium  of  constant  temperature  loses  heat  at  a  rate  proportional 
to  the  difference  in  temperature  between  it  and  the  surrounding 
medium.  This,  then,  is  a  case  of  the  compound  interest  law.  If 
6  denotes  temperature  of  the  cooling  body  above  that  of  the  surround- 
ing medium  at  any  time  t,  we  must  have 

e  =  ae~K 

The  constant  a  must  be  the  value  of  8  when  i  =  0,  or  the  initial  tem- 
perature of  the  body. 

Exercises 

1.  A  thermometer  bulb  initially  at  temperature  19°.3  C.  is  exposed  to 
the  air  and  its  temperature  B  observed  to  be  14°.2  C.  at  the  end  of 
twenty  seconds.  If  the  law  of  cooling  be  given  by  e  =  ffoe"",  where 
t  is  the  time  in  seconds,  find  the  value  of  6  and  6. 

Soltjtion:  The  condition  of  the  problem  gives  9  =  19.3  when  <  =  0, 
hence,  Bo  =  19.3.     Also,  14.2  =  19.3e-206.     This  gives 

log  19.3  -  20b  log  e  =  log  14.2, 

from  which  6  can  be  readily  computed. 

2.  If  IJ  percent  of  the  incident  light  is  lost  when  Ught  is  directed 
through  a  plate  of  glass  0.3  cm.  thick,  how  much  light  would  be  lost  in 
penetrating  a  plate  of  glass  2  cm.  thick?  ■ 

3.  Forty  percent  of  the  incident  light  is  lost  when  passed  through 
a  place  of  glass  2  inches  thick.  Find  the  value  of  a  in  the  equation 
L  =  LoB'"',  where  t  is  thickness  of  the  plate  in  inches,  L  is  the  percent 
of  light  transmitted,  and  Lo  =  100. 

4.  As  I  descend  a  mountain  the  pressure  of  the  air  increases  each 
foot  by  the  amount  due  to  the  weight  of  the  layer  of  air  1  foot  thick. 
As  the  density  of  this  layer  is  itself  proportional  to  the  pressure,  show 
that  the  pressure  as  I  descend  must  increase  by  the  compound  inter- 
est law. 

6.  Power  is  transmitted  in  a  clock  through  a  train  of  gear  wheels 


§154]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  283 

n  in  number.  If  the  loss  of  power  in  each  pair  of  gears  is  10  per- 
cent, draw  a  curve  showing  the  loss  of  power  at  the  nth  gear. 

Note:  The  graphical  method'  of  §121,  Figs.  98,  99,  may  appro- 
priately be  used. 

6.  Given  that  the  intensity  of  light  is  diminished  2  percent  by 
passing  through  one  pane  of  glass,  find  the  intensity  /  of  the  light 
after  passing  through  n  panes. 

7.  The  temperature  of  a  body  cooling  according  to  Newton's  law 
.^fell  from  30°  to  18°  in  six  minutes.     Find  the  equation  connecting 

the  temperature  of  the  body  and  the  time  of  cooling. 

154.  Empirical  Curves  on  Semi-logarithmic  Coordinate  Paper. 

One  of  the  most  important  uses  of  semi-logarithmic  paper  is  in 
determining  the  functional  relation  between  observed  data,  when 
such  data  are  connected  by  a  relation  of  the  exponential  form 
as  already  indicated  in  §160.  Suppose,  for  example,  that  the 
following  are  the  results  of  an  experiment  to  determine  the  law 
connecting  two  variables  x  and  y : 

0.04       0.18       0.36       0.51       0.685       0.833       0.97 


5.3         4.4        3.75       3.1         2.6  2.33         1.9 

If  the  equation  connecting  x  and  y  is  of  the  exponential  form,  the 
points  whose  coordinates  are  given  by  corresponding  values  of  x 
and  y  in  the  table  will  lie  in  a  straight  line  on  semi-logarithmic 
paper,  except  for  such  errors  as  may  be  due  to  inaccuracies  in  the 
observations.  Plotting  the  points  on  semi-logarithmic  coordinate 
paper,  we  find  that  they  lie  nearly  on  the  line  PQ  (Fig.  115). 
Assuming  that,  if  the  data  were  exact,  the  points  would  lie  exactly 
on  this  line,'  we  may  proceed  to  determine  the  equation  of  this  line 
as  approximately  representing  the  relation  between  x  and  y. 
It  is  easy  to  find  the  equation  of  such  a  line  referred  to  the  uni- 

^  We  would  not  be  at  liberty  to  make  such  an  assumption  if  tte  variation  of  the 
points  away  from  the  line  was  of  a  character  similar  to  that  represented  by  the  dots 
near  the  top  of  Fig.  115.  These  points,  although  not  departing  greatly  from  the 
line  shown;  depart  from  it  systematically.  That  is,  they  lie  below  it  at  each  end , 
and  above  it  in  the  center,  seeming  to  approximate  a  curve  (such  as  the  one  shown 
dotted")  more  nearly  than  the  line.  The  points  arranged  about  the  line  PQ  depart 
as  far  from  that  line  as  do  the  points  above  the  higher  line,  but  they  do  not  depart 
systematically,  as  if  tending  to  lie  along  a  smooth  curve.  When  points  arrange 
themselves  as  at  the  top  of  Fig.  115,  one  must  infer  that  the  relation  connecting 
the  given  data  is  not  exponential  in  character. 


284        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§154 

form  scale  AB  and  BC  of  form  M5.  We  may  imagine  that  all 
rulings  are  erased  and  replaced  by  extensions  of  the  uniform  AB 
scale,  as  in  Fig.  111.    The  equation  of  the  line  PQ  is  then 

Y  =  mx  +  B,  (1) 

where  m  is  the  slope,  and  B  is  the  T-intercept.     Now,  for  PQ, 


O-Jlf 


f-. 

iiiiiiiii 

IIIIIIIII 



rjmiiii 

^Vs, 

9: 

s 

8 

"^v 

8  : 

N, 

- 

7 

^<=r 

, 

^V 

C 

^*4 

V 

P 

N, 

*^^[)  : 

^^ 

4 

^^^^ 

4  : 

■~~^v^ 

1 

3 

;^ 

8  " 

^ 

-^ 

1 

^v 

■^^2   = 

1 

1  : 

" 

"" 



A    L      »''         0.2  0.3  0.4  O.S  0.6         0.7  0.6  0.0        1.0B 

Semi  Logarithmic  Paper 

Fig.  115. — Empirical  equations  determined  by  use  of  form  Mb. 

m  =  —  0.447  and  B  =  0.730  =  log  5.37.    Equation  (1)  becomes, 
therefore 

y  =  -  0.447a;  +  log  5.37 
or,  replacing   Y  by  log  y,  in  order  to  refer  the  curve  to  the 
scales  AB  and  LM, 

log  y  -  log  5.39  =  -  0.447a:, 
whence 

y  =  s.ascio-""'*)  (2) 


§155]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  285 

If  it  is  desired  to  express  the  relation  to  the  base  e  instead  of 
base  10,  we  may  note  10  =  e2-''026  (§147,  equation  (8)),  or,  sub- 
stituting in  (2), 

y  =  5.39  (e2-303)-o.447x 

=  5.39  e-i»"«  (3) 

The  same  result  could  have  been  obtained  directly  by  determin- 
ing the  slope  of  PQ  from  the  uniform  scale  AD  at  the  left  of 
Form  Mb. 

155.  Change  of  Scale  on  Semi-logaritbmic  Paper.  A  sheet  of 
semi-logarithmic  paper,  form  Af5,  is  a  square.  If  sheets  of  this 
paper  be  arranged  "checker-board  fashion"  over  the  plane,  then 
the  vertical  non-uniform  scale  will  be  a  repetition  of  the  scale  LM, 
Fig.  115,  except  that  the  successive  segments  of  length  LM  must  be 
numbered  1,  2,  3,  .  ,9  for  the  original  LM,  then  10,  20, 
30,  ,  90  for  the  next  vertical  segment  of  the  checker-board, 

then  100,  200,  300,  ,  900,  for  the  next,  etc.    It  is  obvious, 

therefore,  that  the  initial  point  A  of  a  sheet  of  semi-logarithmic 
paper  may  be  said  to  have  the  ordinate  1,  or  10,  or  100,  etc.,  or 
10~^,  lO"'',  etc.,  as  may  be  most  convenient  for  the  particular 
graph  under  consideration.  The  horizontal  scale  being  a  uniform 
scale,  any  values  of  x  may  be  plotted  to  any  convenient  scale  on 
it,  as  when  using  ordinary  squared  paper.  However,  if  the  hori- 
zontal unit  of  length  (the  length  AB,  form  Mb)  be  taken  as  any 
value  different  from  unity,  then  the  slope  m  of  the  line  PQ  drawn 
on  the  semi-logarithmic  paper  can  only  be  found  by  dividing  its 
apparent  slope  by  the  scale  value  of  the  side  AB.  That  is,  the 
correct  value  of  m  in 

y  —  &10""' 
is,  in  all  cases, 

_  apparent  slope  of  PQ 
scale  value  of  AB 

The  "apparent  slope"  of  PQ  is  to  be  measured  by  applying  any 
convenient  uniform  scale  of  inches,  centimeters,  etc.,  to  the 
horizontal  and  vertical  sides  of  a  right  triangle  of  which  PQ  is  the 
hypotenuse. 


286        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§156 

Exercises 

1.  A  thermometer  bulb  initially  at  temperature  19°.3  C.  is  exposed 
to  the  air  and  its  temperature  e  noted  at  various  times  t  (in  seconds) 
as  follows: 

t  0         20        40      60      80      100     120 


19.3      14.2     10.4     7.6     5.6     4.1     3.0 

Plot  these  results  on  semi-logarithmic  paper  and  test  whether  or  not 
e  follows  the  compound  interest  law.  If  so,  determine  the  value  of 
So  and  6  in  the  equation  6  =  SolO"".  Note  that  the  last  point  given 
by  the  table,  namely  t  =  120,  6  —  3.0,  goes  into  a  new  square  if  the 
scale  AB  be  called  0—100.  If  the  scale  AB  be  called  0—200  then  all 
entries  can  appear  on  a  single  sheet  of  form  Af5. 
2.  Graph  the  following  on  semi-logarithmic  paper: 


n 

\n 

1 

li 

2 

2h 

3 

w 

1.6 

3.0 

5.1 

8.0 

14.0 

23.0 

Show  that  the  equation  connecting  n  and  wis  w  =  lO"'"". 

Sttggestion:  The  scale  AB,  form  Mb,  may  be  called  0 — 5  for  the 
purpose  of  graphing  n. 

3.  Graph  the  following  on  semi-logarithmic  paper,  and  find  the 
equation  connecting  n  and  w. 


n 

0.2 

0.4 

0.6 

0.8 

1.0 

1.2 

1.4 

1.6 

w 

2.60 

3.41 

4.45 

5.75 

7.56 

9.85 

13.0 

16.6 

4.  A  circular  disk  is  suspended  in  a  horizontal  plane  by  a  fine  wire 
at  its  center.  When  at  rest  the  upper  end  of  the  wire  is  turned  by 
means  of  a  supporting  knob  through  30°  The  successive  angles  of 
the  torsional  swings  of  the  disk  from  the  neutral  point  are  then  read 
at  the  end  of  each  swing  as  follows: 

Swing  number 


1 

■     2 

3 

4 

6 

6 

7 

26°.4 

23°.2 

20°.5 

18°.0 

15°.9 

14°.0 

12°.3 

Angle 

Show  that  the  angle  of  the  successive  swings  follows  the  compound 
interest  law  and  find  in  at  least  two  different  ways  the  equation  con- 
necting the  number  of  the  swing  and  the  angle.  Show  also  by  the 
slide  rule  that  the  compound  interest  law  holds.     [tt>  -  SOlO"-"""] 

166.  The  Power  Function  Compared  with  the  Exponential^ 
Function.     It  has  been  emphasized  in  this  book  that  the  funds- 


§156]  LOGARITHMIC  AND  EXPONENTIAL  FIJNCTIONS  287 

mental  laws  of  natural  science  are  three  in  number,  namely:  (1) 
the  parabolic  law,  expressed  by  the  power  function  y  =  ax" 
where  n  may  be  either  positive  or  negative;  (2)  the  harmonic  or 
periodic  law,  y  =  asin  nx,  which  is  fundamental  to  all  periodically 
occurring  phenomena;  and  (3)  the  compound  interest  law  dis- 
cussed in  this  chapter.  While  there  are  other  important  laws  and 
functions  in  mathematics,  they  are  secondary  to  those  expressed 
by  these  fundamental  functions.  The  second  of  the  functions 
above  named  wUl  be  more  fully  discussed  in  the  chapter  on  waves. 
The  discussion  of  the  compound  interest  law  should  not  be  closed 
without  a  careful  comparison  of  power  functions  and  exponential 
functions. 
The  characteristic  property  of  the  power  function 

y  =  ax"  (1) 

is  that  as  x  changes  hy  a  constant  factor,  y  changes  by  a  constant 
factor  also.     Let 

y  =  ax"  =  f(x).  (2) 

Let  X  change  by  a  constant  factor  m,  so  that  the  new  value  of  x 
is  mx.    Call  y'  the  new  value  of  y.     Then 

y'  =  a{mx)"  =  f{mx).  (3) 

That  is, 

y'       a(mx)"  ,., 

—  =  -^^ — '-  =  m",  (4) 

y  ax" 

which  shows  that  the  ratio  of  the  two  y's  is  independent  of  the  value 
of  X  used,  or  is  constant  for  constant  values  of  m. 

Another  statement  of  the  law  of  the  power  function  is:  As  .t 
increases  in  geometrical  progression,  y,  or  the  power  function,  in- 
creases in  geometrical  progression  also. 

r 
Let  m  be  nearly  1,  say  1  +  t^,  where  r  is  the  percent  change  in  x 

and  is  small,  then  we  have 


y'    K^'  +  m)     '^K^)' 


-  =     '    ,,  ;""'    =  ^ ^^^^^  =  (1  +  r)"  =F  1  +  nr        (5) 

y  fix)  ax" 

by  the  approximation  formula  for  the  binomial  theorem  (§113). 


288        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§166 
Hence, 


d'  +  m)-^^^^ 


y  fix)  100 


(6) 


f^)=nr.  (7) 


100 

y 

The  left-hand  member  is  the  percent  change  in  y  or  infix).  The 
number  r  is  the  percent  change  in  the  variable  a;.  Therefore 
(7)  states  that  for  small  changes  of  the  variable  the  percent  of 
change  in  the  function  is  n  times  the  percent  of  change  in  the  variable. 
Let  the  exponential  function  be  represented  by 

y  =  ae''  =  Fix).  (8) 

As  already  noted  in  the  preceding  sections,  increasing  x  by  a  con- 
stant term  increases  y,  or  the  function,  by  a  constani  factor.    Thus 

y'      F{x  +  h)      aeoi'^")  ' 

y  Fix)      ~     oe»'     ~      '  ^  ' 

which  is  independent  of  the  value  of  x,  or  is  constant  for  constant  h. 
The  expression  e'*  is  the  factor  by  which  y  or,  the  function,  is  in- 
creased when  X  is  increased  by  the  term,  or  increment,  h.  See 
§147. 

In  other  words,  as  x  increases  in  arithmetical  progression,  y, 
or  the  exponential  function,  increases  in  geometrical  progression. 

The  percent  of  change  is 

[Fix  +  /i)  -  Fix)- 


100 


^  =  100  [e'*  -  1],  (10) 


Fix) 

which  is  constant  for  constant  increments  h  added  to  the  variable  x. 

If  X  change  by  a  constant  percent  from  a;  to  a;  ( 1  +  t?^)  ,  it  will 

be  found  that  the  percent  change  in  the  function  is  not  constant, 
but  is  variable. 

The  above  properties  enable  one  to  determine  whether  measure- 
ments taken  in  the  laboratory  can  be  expressed  by  functions  of 
either  of  the  types  discussed;  if  the  numerical  data  satisfy  the 
test  that  if  the  argument  change  by  a  constant  factor  the  function 
also  changes  by  a  constant  factor,  then  the  relation  may  be  repre- 
sented by  a  power  function.    If,  however,  it  is  found  that  a  change 


§157]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  289 

of  the  argument  by  a  constant  increment  changes  the  function 
by  a  constant  factor,  then  the  relation  can  be  expressed  by  an 
equation  of  the  exponential  type. 

We  have  already  shown  how  to  determine  the  constants  of  the 
exponential  equation  by  graphing  the  data  upon  semi-logarithmic 
paper.  In  case  the  equation  representing  the  function  is  of  the 
form 

y  =  ae^''  +  c,  (11) 

then  the  curve  is  not  a  straight  line  upon  semi-logarithmic  paper. 
If  tabulated  observations  satisfy  the  condition  that  the  function 
less  (or  plus)  a  certain  constant  increases  by  a  constant  factor  as 
the  argument  increases  by  a  constant  term,  then  the  equation  of 
the  type  (11)  represents  the  function  and  the  other  constants  can 
readily  be  determined. 

The  determination  of  the  equations  of  curves  of  the  parabolic 
and  hyperbolic  type  is  best  made  by  plotting  the  observed  data 
upon  logarithmic  coordinate  paper  as  explained  in  the  next  section. 

157.  Logarithmic  Coordinate  Paper.  If  coordinate  paper  be 
prepared  on  which  the  uniform  x  and  y  scales  are  both  replaced 
by  non-uniform  scales  divided  proportionately  to  log  x  and  log  y, 
respectively,  then  it  is  possible  to  show  that  any  curve  of  the  para- 
bolic or  hyperbolic  type  when  drawn  upon  such  coordinate  paper  will 
be  a  straight  line.  This  kind  of  squared  paper  is  called  logarithmic 
paper,  and  is  illustrated  in  Fig.  116. 

To  find  the  equation  of  a  line  PQ  on  such  paper,  we  imagine,  as 
in  the  case  of  semi-logarithmic  paper,  that  aU  rulings  are  erased 
and  replaced  by  continuations  of  the  uniform  scales  ON  and  MN, 
on  which  the  length  ON  or  MN  is  taken  as  unity.  Denoting,  as 
before,  distances  referred  to  these  uniform  scales  by  capital  letters, 
we  may  write  as  the  general  equation  of  a  straight  line 

Y  =  mX  +  B.  (1) 

In  the  case  of  the  line  PQ,  m  =  0.505,  B  =  0.219,  and  hence 

Y  =  0.505X  +  0.219. 

But,    Y  =  log  y,  X  =  log  X,  where  y  and  x  represent  distances 

19 


290        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§157 

measured  on  the  scale  LM  and  LO  respectively,  and  0.219  = 
log  1.65.    Hence 

log  y  =  0.505  log  X  +  log  1.65 
or 

log  y  —  log  1.65  =  0.505  log  .r. 


0  ar   1 


10  AT 


1 





10 

u 

Q 

^ 

^ 

^ 

^ 

y 

y 

^ 

^ 

^ 

y 

5 

^ 

^ 

^ 

4 

p 

N 

: 

2 

L 

1 
»0 

1  2  3  4567S9     10 

Single  LogarLtlimic,  Scale  of  Oommon  LoearUlims  In  Margins 

Fig.  116. — Logarithmic    coordinate    paper,    form    itf4.     The    finer 
rulings  of  form  Af 4  are  not  reproduced. 


This  may  be  written  in  the  form 

logj^  =  log  a;" 


whence 


or 


y       —  T.0.505 

1.65  ~  ""       • 


y  =  1.65a;<'-=''^ 


(2) 


I 
§157]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   291 

In  general,  if  £  =  log  6,  we  may  write  the  equation  (1)  in  the 
form 

y  —  bx'"  (3) 

If  the  straight  line  on  logarithmic  paper  passes  through  the 
point  (1,  1)  its  Cartesian  equation  is 

Y  =  mX,  (4) 

or,  referred  to  the  logarithmic  scales, 

log  y  =  m  log  X  —  log  a;"", 
or 

y  =  X".  (5) 

If  the  straight  line  on  logarithmic  paper  passes  through  the  point 
(a,  6)  with  slope  m,  its  equation  referred  to  the  logarithmic  scales 
is 

(6) 


I  =  [;]■ 


On  logarithmic  paper,  form  Mi,  the  numbers  printed  in  the 
lower  and  in  the  left  margin  refer  to  the  non-uniform  scale  in  the 
body  of  the  paper.  By  calling  the  left-hand  lower  corner  the 
point  (1,10),  (10, 10),  (10, 1),  (10, 100),  (1,100)  or  (100, 100),  .  .  .  , 
instead  of  (1,1),  these  numbers  may  be  changed  to  10,  20,  30, 
.  ,  or  to  100,  200,  300,   .    .    .  ,  etc. 

If  the  range  of  any  variable  is  to  extend  beyond  any  of  the  single 
decimal  intervals,  1—10,  10—100,  100—1000,  .  .  .  ,  the  "multiple 
paper,"  form  MQ,  may  be  used,  or  several  straight  lines  may  be  drawn 
across  form  JW4  corresponding  to  the  value  of  the  function  in  each 
decimal  interval,  1 — 10,  10 — 100,  .  .  .,  so  that  as  many  straight 
lines  will  be  required  to  represent  the  function  on  the  first  sheet  as 
there  are  intervals  of  the  decimal  scale  to  be  represented.  However, 
if  the  exponent  m  in  i/  =  bx"  be  a  rational  number,  say  n/r,  then  the 
lines  required  for  all  decimal  intervals  will  reduce  to  r  different  straight 
lines. 

One  of  the  most  important  uses  of  logarithmic  paper  is  the  de- 
termination of  the  equation  of  a  curve  satisfied  by  laboratory 
data.  If  such  data,  when  plotted  on  logarithmic  paper,  give 
a  straight  line,  an  equation  of  the  form  (6)  above  satisfies  the 
observations  and  the  equation  is  readily  found.  The  exponent 
m  is  determined  by  measuring  the  slope  of  the  line  with  an  ordinary 


292         ELEMENTARY  MATHEMATIQAL  ANALYSIS      [§157 


uniform  scale.  The  equation  of  the  line  is  best  found  by  noting 
the  coordinates  of  any  one  point  (o,  6)  and  substituting  these 
and  the  slope  m  in  equation  (6). 

Illustration  1.  Construct  the  semi-cubical  parabola  y=  2x1 
on  logarithmic  paper. 

The  result  is  a  straight  line  of  slope  -|  cutting  the  line  LM,  Fig.  116, 
at  the  point  marked  2. 

100? ^  ■»  9    ~     I 

90 
80 
70 
60 
EO 

40 
30 


20 


£__________.£/ E 

/  D  F 


2        3     4    5  6  78910  20       30    40  5060 

K  ?  M 

Fig.  117. — Multiple  logarithmic  paper. 


100 


Illustbation  2.    Find  an  empirical  equation  connecting  the  x  and 
y  of  the  accompanying  table. 


X 

y 

X 

y 

5 

1.0 

20 

16.4 

7 

2.0 

30 

37.0 

9 

3.3 

40 

65.0 

15 

9.2 

50 

100.0 

These  points  are  shown  plotted  on  the  multiple  logarithmic  paper 
in  Fig.  117,  as  the  line  PQ.    The  slope  of  this  line  is  found  to  be  2. 


§157]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   293 


Substituting  for  a  and  6  in  (fi)  the  coSrdinates  of  any  of  the  points 
on  the  line,  for  example  (5,  1),  we  get 


y.  =  M  ' 

1       \6/ 


or  2/  =  25^- 


Exercises 

Draw  the  following  on  single  or  multiple  logarithmic  paper,  forms 
M4or  Af6: 


1.  y  =  x,y  =2x,y  =3x,y  =  ix.      i.  y  =  x^^,  y  =  x^^, 


■.^ 


2.  y  =  X,  y  =  x^,  y  =  x\  y  ^  x*.       5.  y  =  2x',  y  =  Ja;',  A  =  itrK 
Z.  y  =  1/x,  y  =  l/x\  y  =  1/xK 

6.  Find  the  empirical  equation  connecting  x  and  y  of  the  following 
table. 


X 

y 

X 

y 

1.6 

3.05 

6.5 

6.40 

2.5 

3.92 

7.5 

6.85 

3.5 

4.65 

8.5 

7.25 

4.5 

5.30 

9.5 

7.70 

5.5 

5.82 

7.  Find  the  empirical  equation  connecting  x  and  y  of  the  following 
table. 


X 

y 

X 

y 

1.2 

2.15 

2.0 

5.90 

1.3 

2.50 

2.3 

7.80 

1.5 

3.85 

2.5 

9.30 

1.7 

4.30 

8.  Find  the  empirical  equation  connecting  x  and  y  of  the  following 
table. 


X 

y 

X 

y 

1.5 

10.0 

4.5 

3.30 

2.0 

7.5 

5.0 

2.98 

2.5 

6.0 

6.0 

2.49 

3.0 

5.0 

7.0 

2.12 

3.5 

4.25 

8.0 

1.87 

4.0 

3.73 

9.0 

1.65 

294         ELEMENTARY  MATHEMATICAL  ANALYSIS     [§157 


Draw  the  following  on  single  or  multiple  logarithmic  paper  as  best 
suits  the  particular  example.  Carefully,  label  the  scales  and  indicate 
the  true  numerical  value  of  the  division  points.  Use  common  sense 
values  of  the  variables — ^for  example  in  exercise  16  do  not  graph  for 
speed  over  30  knots. 

9.  p  =  0.003«^,  where  p  is  the  pressure  in  pounds  per  square  foot 
on  a  flat  surface  exposed  to  a  wind  velocity  of  v  miles  per  hour. 

Suggestion:  The  "common  sense"  range  for  v  is  from  w  =  10  to 
V  =  100. 


c 

'         1     G, 

.2     .; 

1 

,' 

F 

.5 

r 

.6 

r 

.8 

1 

9 

iB 

f 

9 

\ 

\ 

/ 

8 

/ 

/ 

7 

\( 

A 

6 

f      V 

/ 

\^ 

5 

/ 

\ 

/ 

\ 

/ 

^ 

<, 

/ 

^ 

K 

3 

Sj 

-.5 

> 

\ 

n 

s 

N 

Iv 

.      o 

E 
1 

\-.2 

C 

H 

-.1 

A 

1 

i 

I 

a 

< 

i 

5 

r 

8     9 

lOo 

Fig.  118.— Diagram  for  Exercise  10. 

10.  Find  the  equ^,tions  in  rectangular  coordinates  of  the  lines  EF 
and  GH  of  Fig^  118. 

11.  V  =  c-\/rs  for  c  =  110  and  r  =  1. 

12.  /  =  y/2gh  for  g  =  32.2. 

13.  C  =  E/B  where  E  =  110  volts. 

14.  s  =  Igt^  where  g  =  32.2. 

16.  T  =  TrVZ/g,  where  g  =  32.2. 

16.  p/po  =  (p/po)^*"',  where  po  =  0.075,  the  weight  of  1  cubic 
foot  of  air  in  pounds  at  70°  F.  and  at  pressure  po  of  14.7  pounds  per 
square  inch. 


§158]   LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   295 

17.  H  =  — p— ,  for  D  =  5000,  10,000,  15,000,  20,000,  where  C  = 
225,  D  is  displacement  in  tons  and  /S  is  speed  in  knots. 

18.  H  =  -go",  for  N  =  100,  200,  300,  400,  500,  600,  700,  800,  900, 

1000.     d  is  the  diameter  of  cold  rolled  shafting  in  inches.    The  line 
should  be  graphed  for  values  of  d  between  d  =  1  and  d  =  10. 

19.  F  =  O.OOOSilWBN',  where  N  is  revolutions  per  minute,  R  is 
radius  in  feet,  W  is  weight  in  pounds,  and  F  is  centrifugal  force  in 
pounds. 


'• 

/'^ 

;    ;.::;:p 

' 

y 

1 

0.9 
0.8 

= 

E 

— 

-7 

2 

'',' 

;';■:  ::* 

...... 

;:  :  :       : 

^ 

T 

0.0 

^ 

y 

;2 

^^- 

;':';;   ; 

12 

.' 

1  Sj  . 

.' 

.    ^^ 

c'  t'  ■'    % 

0.R 

i 

■=;-::^ 

=  !■■ 

^ 

;=-: 

="^:^  is 

.::. 

E 

~  7.^ 

"2*^ 

-  Ji. :    .  J^  ^ 

— 

^^> 

;^::   :    : 

-■^. 

,  • 

^ 

<-ii 

> 

< 

<' 

0.1 

a09 

: 

: 

1 

y 

;;;::  ;:; 

',-; 

1 

1 

i:::  :      _: 

= 

0.07 

■p 

2 

z  r  = 

-^      ,. ,.    z 

- 

i'!  ::!I  - 

0.04 

X-'' 

2^:   :.  I_ 

« 

a2    0.3    a4 


£=I*iigthofCreBtinJBet 
^=Heaa  on  Creat  in  Feet 
g  =BlB0liai^e  In  Second  Feet 


ae   0.8  1.0  3  3       4     6    6  7  8  910 

*      0.7  0.9 
Discharge  over  Trapezoidal  Wiex 


Fig.  119. — A  weir  formula  graphed  on  multiple  logarithmic  pa  per. 

20.  g  =  3.37M^  for  L  =  0.1,  0.2,  0.3,  0.4,  0.5,  0.6,  0.7,  0.8,  0.9, 
1.0.     See   Fig.    119. 

O.SSF'*^ 

21.  H  =  — jj-yjj — ,  where  V  is  the  velocity  of  water  in  feet  per 

second  under  the  head  of  H  feet  per  10,000  feet  in  clean  cast-iron  pipe 
of  diameter  d  feet.     See  Fig.  120. 

158.  Slims  of  Exponential  Functions.  Functions  consisting  of 
the  sum  of  two  different  exponential  functions  are  of  frequent 
occurrence  in  the  application  of  mathematics,  especially  in  elec- 
trical science.  Types  of  fundamental  importance  are  e"  +  e"" 
and  e»  —  e-"  which  are  so  important  that  the  forms  (e"  +  e~")/2 
and  (e"  —  e~")/2  have  been  given  special  names  and  tables  of 


296        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§158 

their  values  have  been  computed  and  printed.    The  first  of 

these  is  called  the  hyperbolic  cosine  of  u  and  the  second  is  called 

the  hyperbolic  sine  of  u ;  they  are  written  in  the  following  notation : 

cosh  w  =  (6«  +  e~")/2,     sinh  li  =  (e«  —  e~")/2. 


Triction  Head  in  Feei  per  IDOO  Ft. of  Pipe 


Not«: 

For  opoD  ooadults,  multlplj  IlydraUllo  Radlut  bj  4  to 
get  Equivalent  SUmetsT.  Diagram  givu  noarly  Boms 
reaults  aa  KuttWB  Fonnulm  Kith  n=.011. 
Fur  old  or  foul  pipes  multlplj  required  head  bj  1.4& 
f>00  ^  ^^^  o'  divide  diagram  veloolt;  b;  I>20  to  1,28  for 
V=  2  to  &  feet  per  Beoond .  j 

g    For  Bubb  pipes  ffsO-fiO^lia 


Diagram,  of  Flow  In  Clean  Oaat  Iron  or  Wrought  Iron  Pipes 
Baaed  on  the  Formula,  H,  in  Feet  per  1000  Feet  =  0,38  K"^- 

FiG.  120. — A  compHcated  example  of  the  use  of  multiple  logarithmic  paper,  Form 
MQ.     From  Transactions  Am.  Soc.  C.  E.,  Vol.  LI. 

If  X  =  a  cosh  u  and  y=  a  sinh  u,  then  squaring  and  subtracting 


x^  —  y^  =  a2(cosh''  u  —  sinh^  u) 


=..p 


+  2  +  e-2" 


2  +  e--' 


4  4 

Therefore  the  hyperbolic  functions 

x=  a  cosh  u,  and  y  =  a  sinh  u 


'1 


§158]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  297 
appear  in  the  parametric  equations  of  a  rectangular  hyperbola 

just  as  the  circular  functions 

X  =  a  cos  B,  and  y  =  asind 
appear  in  the  parametric  equations  of  the  circle 

a;2  -)-  j/2   =  o^ 


4 

1 

L 

3.6 

yi 

— 

\ 

3 

\ 

/ 

\ 

2.5 

w 

' 

I 

2 

f 

*^ 

1,5 

^ 

^1 

^ 

V' 

V 

•  y 

e" 

N 

\5 

l/ 

V 

-yie- 

■ 



X 

^ 

1 — ' 

^ 

-3.5 

-3 

-2. 

-2 

-1. 

-1 

H 

0 

6 

1 

1.5 

2 

2.S 

3 

3.5 

/ 

/ 

l.S 

1 

-2 

/ 

2.5 

/ 

-3 

/ 

3.5 

/ 

-4 

4.5 

-5 

Fig.  121. — The  curves  of  the  hyperboUc  sine  and  cosine. 

The  graphs  of  y  =  a  cosh  x  and  y  =  a  sinh  x  were  called  for  in 
exercises  1,  2,  §146.  They  are  shown  in  Fig.  121.  The  first 
of  these  curves  is  formed  when  a  chain  is  suspended  between  two 
points  of  support;  it  is  called  the  catenary.  These  two  curves 
are  best  drawn  by  averaging  the  ordinates  oi  y  =  e'  and  y  —e~', 
and  the  ordinates  oi  y  =  e''  and  y  =  —  e"'. 

Curves  whose  equations  are  of  the  form 

y  =  ae""*  +  be"' 

take  on  quite  a  variety  of  forms  for  various  values  of  the  constants. 
A  good  idea  of  certain  important  types  can  be  had  by  a  comparison 
of  the  curves  of  Figs.  122  and  123. 


298        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§158 


1.& 

(1) 

I 

.75 

\w 

(1)  2^=e-*+o,Be-"'* 

(2)  3/=e""        .JO, 

(4)  y=e:Z-ojseJ"' 
(6)2/=e"-i.Be  *°* 

bA(s) 

w 

5 

P 

\ 

- 

25 

m 

\ 

N^ 

^^ 

— ^ 

1  1.5 

Pig.  123. 


2.6 


§159]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  299 

The  student  should  arrange  in  tabular  form  the  necessary 
numerical  work  for  the  construction  of  the  curves  of  Figs.  122 
and  123. 

If  the  coefficient  of  the  second  exponent  be  increased  in  absolute 
value,  the  points  of  intersection  with  the  F-axis  remain  the  same, 
but  the  region  of  close  approach  of  the  curves  to  each  other  is 
moved  along  the  curve  y  =  e-'  to  a  point  much  nearer  the  Y-axis 
as  can  be  seen  by  comparing  Fig.  123  with  Fig.  122. 

159.  *Damped  Vibrations.  If  a  body  vibrates  in  a  medium  like 
a  gas  or  liquid,  the  amplitude  of  the  swings  are  found  to  get  smaller 
and  smaller,  or  the  motion  slowly  (or  rapidly  in  some  cases)  dies  out. 
In  the  case  of  a  pendulum  vibrating  in  oil,  the  rate  of  decay  of  the 
amplitude  of  the  swings  is  rapid,  but  the  ordinary  rate  of  the  decay  of 
such  vibrations  in  air  is  quite  slow.  The  ratio  between  the  lengths 
of  the  successive  amplitudes  of  vibration  is  called  the  damping  factor 
or  the  modulus  of  decay. 

The  same  fact  is  noted  in  case  the  vibrations  are  the  torsional 
vibrations  of  a  body  suspended  by  a  fine  wire  or  thread.  Thus  a 
viscometer,  an  instrument  used  for  determining  the  viscosity  of 
lubricating  oils,  provides  means  for  determining  the  rate  of  the  decay 
of  the  torsional  vibration  of  a  disk,  or  of  a  circular  cylinder  suspended 
in  the  oil  by  a  fine  wire.  The  "amplitude  of  swing"  is  in  this  case  the 
angle  through  which  the  disk  or  cylinder  turns,  measured  from  its 
neutral  position  to  the  end  of  each  swing. 

In  all  such  cases  it  is  found  that  the  logarithms  of  the  successive 
amplitudes  of  the  swings  differ  by  a  certain  constant  amount  or,  as 
it  is  said,  the  logarithmic  decrement  is  constant.  Therefore  the 
amplitudes  must  satisfy  an  equation  of  the  form 

A  =  ae~^ 

where  A  is  amplitude  and  /  is  time.     The  actual  motion  is  given  by  an 
equation  of  the  form 

y  =  ae~^  sin  ct, 

A  study  of  oscillations  of  this  type  will  be  taken  up  more  fully  in 
the  calculus.  For  the  present  it  will  suffice  to  graph  a  few  examples. 
Let  the  expression  be 

y  =  g-f/B  sin  t.  (1) 

A  table  of  values  of  t  and  y  must  first  be  derived.     There  are  three 
ways  of  proceeding;  (1)  Assign  successive  values  to  t  urespective  of 


300         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§169 

the  period  of  the  eine  (see  Table  V  and  Fig.  124).  (2)  Select  for  the 
values  of  t  those  values  that  give  aliquot  parts  of  the  period  2t  of  the 
sine  (see  Table  VI  and  Kg.  125).  (3)  Draw  the  sinusoid  y  =  sin  t 
carefuUy  to  scale  by  the  method  of  §56;  then  draw  upon  the  same 


V 

/ 

'^ 

s 

. 

U>b 

/ 

\ 

\ 

v'^ 

A 

r 

\ 

^^1,0     1 

1     12  4" 

t 

U 

> 

V 

[^ 

t 

■      J 

'i 

-' 

■^13 

0.B 

\ 

/ 

1.6 
V 
\ 


-1.6 


Fig.  124. — The  curve  y  =  e"*/'  sin  t. 


\ 

s*. 

/ 

N 

>N 

t" 

/ 

\ 

<1 

r^ 

\ 

TT 

ITT 

/^ 

'n 

lir 

4T 

J 

( 

V 

i  1 

[/ 

2     1 

4      1 

6J 

M 

U2 

^ 

4     2 

B     2 

\ 

>^ 

^ 

^ 

rr"' 

/ 

Fig.  125. — The  curve  y  =  e"*/'  sin  t. 


coordinate  axes,  using  the  same  units  of  measure  adopted  for  the  sinus- 
oid, the  exponential  curve  y  =  e~'/^;  finally  multiply  together,  on  the 
slide  rule,  corresponding  ordinates  taken  from  the  two  curves,  and 
locate  the  points  thus  determined. 


§159]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS    301 


The  first  method  involves  very  much  more  work  than  the  second 
for  two  principal  reasons:  First,  tables  of  the  logarithms  of  the 
trigonometric  functions  with  the  radian  and  the  decimal  divisions 
of  the  radian  as  argument  are  not  available;  for  this  reason  57.3° 
must  be  multiplied  by  the  value  of  t  in  each  case  so  that  an  ordinary 
trigonometric  table  may  be  used;  second,  each  of  the  values  written 
in  column  (3)  of  the  table  must  be  separately  determined,  while  if 
the  periodic  character  of  the  sine  be  taken  advantage  of,  the  numerical 
values  would  be  the  same  in  each  quadrant. 

TABLE  V 


Table  of  the  function  y 

=  e  "^  sin  t 

1 

2 

3 

4 

5 

t  in  radians 

log  e-'/»  = 
-  (0.0869)i 

log  sin  (  or  log 

sin  57.34  if  ( is 

in  degrees 

logy 

V 

0.0 

-0.0000 

+  0.000 

0.5 

-  0.0434 

9.6807 

9.6372 

+  0.434 

1.0 

-  0.0869 

9.9250 

9.8381 

+  0.689 

1.5 

-  0.1303 

9.9989  ' 

9.8686 

+  0.739 

2.0 

-0.1737 

9.9587 

9.7850 

+  0.610 

2.5 

-  0.2172 

9.7771 

9.5599 

+  0.363 

3.0 

-  0.2606 

9.1498 

8.8892 

+  0.077 

3.5 

-  0.3040 

9.5450 

9.2410 

-0.174 

4.0 

-0.3474 

9.8790 

9.5312 

-  0.340 

4.5 

-0.3909 

9.9901 

9.5992 

-  0.397 

5.0 

-  0.4343 

9.9818 

9.5475 

-  0.353 

5.5 

-  0.4777 

9.8485 

9.3708 

-  0.235 

6.0 

-  0.5212 

9.4464 

8.9252 

-  0.084 

6.5 

-  0.5646 

9.3322 

8.7679 

+  0.059 

7.0 

-  0.6080 

9.8175 

9.2095 

+  0.162 

7.5 

-  0.6515 

9.9722 

9.3207 

+  0.209 

8.0 

-  0.6949 

9.9954 

9.3005 

+  0.200 

8.5 

-  0.7383 

9.9022 

9.1634 

+  0.146 

9.0 

-  0.7817 

9.6149 

8.8332 

+  0.068 

9.5 

-  0.8252 

8.8760 

8.0508 

-0.011 

10.0 

-  0.8686 

9.7356 

8.8670 

-0.074 

10.5 

-  0.9120 

9.9443 

9.0323 

-  0. 108 

11.0 

-  0.9555 

9.9999 

9.0444 

-0.111 

11.5 

-  0.9989 

9.9422 

8.9433 

-  0.088 

12.0 

-  1.0424 

9.7296 

8.6872 

-0.049 

The  second  method,  because  of  the  use  of  aliquot  divisions  of  the 


302         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§159 


period  of  tl^e  Bine,  such  as  ir/6  or  ir/12  or  r/l&  or  5r/20,  etc.,  possesses 
the  advantage  that  the  values  used  in  column  (3)  need  be  found  for  one 
quadrant  only  and  the  values  required  in  column  (2)  are  quite  as 
readily  found  on  the  slide  rule  as  in  the  first  method. 


TABLE  VI 
Table  of  the  function  y 


=  e""/*  sin  t 


1 

2 

3 

4 

5 

n  =  t  in 
units  of 

ir/6  radians 

log  e- »"■/'"  = 
-  (0.0455)» 

log,  ain  nw/6 

log!/ 

y 

0 

1 

-0.0000 
-  0.0455 

0.000 
+  0.450 

9.6990 

9.6535 

2 

-0.0910 

9.9375 

9.8465 

+  0.702 

3 

-  0.1364 

0.0000 

9.8636 

+  0.731 

4 

-  0.1819 

9.9375 

9.7556 

+  0.570 

5 

-  0.2274 

9.6990 

9.4716 

+  0.296 

6 

-0.2729 

+  0.000 

7 

-  0.3184 

9. ,6990 

9.3806 

-  0.240 

8 

-  0.3638 

9.*9375 

9.5737 

-  0.375     _ 

9 

-  0.4093 

0.0000 

9.5907 

-0.390 

10 

-0.4548 

9.9375 

9.4827 

-  0.304 

11 

-  0.5003 

9.6990 

9.1987 

-0.158 

12 
13 

-  0.5458 
-0.5912 

0.000 
+  0.128 

9.6990 

9.1078 

14 

-  0.6367 

9.9375 

9.3008 

+  0.200 

15 

-  0.6822 

0.0000 

9.3178 

+  0.208 

16 

-0.7277 

9.9375 

9.2098 

+  0.162 

17 

-0.7732 

9.6990 

8.9258 

+  0.084 

18 
19 

-  0.8186 
-0.8641 

0.000 
-  0.068 

9.6990 

8.8349 

20 

-  0.9016 

9.9375 

9.0279 

-0.107 

21 

-  0.9551 

0.0000 

9.0449 

-  0.111 

22 

-  1.0006 

9.9375 

8.9369 

-0.087 

23 

-  1.0460 

9.6990 

8.6530 

-0.045 

24 

-  1.0915 

0.000 

The  third  method  is  perhaps  more  desirable  than  either  of  the 
others  if  greater  than  two  figures  accuracy  is  not  required.  The 
curve  can  readily  be  drawn  with  the  scale  units  the  same  in  both 
dimensions,  as  is  sometimes  highly  desirable  in  scientific  applications. 

In  Figs,  124  a^d  125  a,  larger  unjt  ha?  be^n  used  on  the  vertical 


§159]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  303 

scale  than  on  the  horizontal  scale.  In  Fig.  125  the  horizontal  unit 
is  incommensurable  with  the  vertical  unit.  To  draw  the  curve  to  a 
true  scale  in  both  dimensions  it  is  preferable  to  lay  off  the  coSrdinates 
on  plain  drawing  paper  and  not  on  ordinary  squared  paper.  Rec- 
tangular coordinate  paper  is  not  adapted  to  the  proper  construction 
and  discussion  of  the  sinusoid,  or  of  curves,  like  the  present  one,  that 
are  derived  therefrom. 

Curves  whose  equations  are  of  the  form  y  =  je"*/'  sin  tor  y  = 
Se"'.''  sin  t,  etc.,  are  readily  constructed,  since  the  constants  i,  3,  etc., 
merely  multiply  the  ordinates  of  (1)  by  \,  3,  etc.,  as  the  case  may  be. 
Likewise  the  curve  y  =  e~'*  sin  ex  is  readily  drawn  since  sin  ex  can  be 
made  from  sin  x  by  multiplying  all  abscissas  of  sin  x  by  1/c. 


CHAPTER  X 


TRIGONOMETRIC  EQUATIONS  AND  THE  SOLUTION 
OF  TRIANGLES 

A.  FURTHER  TRIGONOMETRIC  IDENTITIES 

160.  The  circle  p  =  a  cos  d  +  h  sin  0.  In  §74  an  analytical 
proof  was  given  of  the  fact  that  p  =  a  cos  fl  +  6  sin  5  is  the  polar 
equation  of  a  circle  passing  through  the  pole  0  and  having  its 
center  at  the  point  (\a,  56).  The  demonstration  there  given 
should  now  be  reviewed. 
Geometrical  Explanation.  The  following  geometrical  dis- 
cussion should  give  the 
student  a  better  under- 
standing of  the  important 
theorem  of  §74. 

We  know  (§66)  that  pi 
=  a  cos  B  is  the  polar  equa- 
tion of  a  circle  of  diameter 
a,  the  diameter  coinciding 
in  direction  with  the  polar 
axis  OX;  for  example,  the 
circle  OA,  Fig.  126.  Like- 
wise, p2  =  6  sin  0  is  a  circle 
whose  diameter  is  of  length 
6  and  makes  an  angle  of 
-|-90°  with  the  polar  axis 
OX,  for  example,  the  circle 
OB,  Fig.  126.  Also,  p  = 
c  cos  {0  —  6)  is  a  circle  whose 
diameter  c  has  the  direction  angle  5.  See  Theorem  XIV  on  Loci, 
§70.  We  shall  show  that  if  the  radius  vectors  corresponding  to 
any  value  old  in  the  equations  pi  =  a  cos  d  and  pa  =  6  sin  6  be  added 
together  to.make  a  new  radius  vector  p,  then,  for  all  values  of  B, 

304 


Pig.  126. — Combination  of  the  cir- 
cles p  =  a  cos  6  and  p  =  6  sin  9  into  a 
single  circle  p  =  a  cos  B  +  6  sin  e. 


§160]  TRIGONOMETRIC  EQUATIONS  305 

the  extremity  of  p  lies  on  a  circle  (the  circle  OC,  Fig.  126)  of  di- 
ameter Vo^  +  h'^.    In  other  words  we  shall  show  geometrically  that 

p  =  a  cos  &  +  6  sin  fl  (1) 

is  the  equation  of  a  circle. 

In  Fig.  126,  pi  =  a  cos  Q  wiU  be  called  the  a-cirde  OA;  p^  = 
6  sin  6  will  be  called  the  b-circle  OB.  For  any  value  of  the  angle 
6  draw  radius  vectors  OM,  ON,  meeting  the  a-  and  6-circles  respec- 
tively at  the  points  M  and  N.  If  P  be  the  point  of  intersec- 
tion of  MN  produced  with  the  circle  whose  diameter  is  the  diagonal 
OC  of  the  rectangle  described  on  OA  and  OB,  we  shaU  show  that 
OM  +  ON  =  OP,  no  matter  in  what  direction  OP  be  drawn. 

Let  the  circle  last  mentioned  be  drawn,  and  project  BC  on  OP. 
Since  ONB  and  OPC  are  right  angles,  NP  is  the  projection  of 
BC  {=  a)  upon  OP.  But  OM  also  is  the  projection  of  a  (=  OA) 
upon  OP.  Hence  NP  =  OM  because  the  projections  of  equal 
parallel  lines  on  the  same  line  are  equal.  Therefore,  for  all  values 
of  d,  NP  =  pi  and  OP  =  ON  +  NP  =  pi  +  pi,  which  is  the  fact 
that  was  to  be  proved. 

Designating  the  angle  AOC  by  5,  the  equation  of  the  circle  OC  is 

by  §70.  

P  =  Va^  4-  h'^  cos  {6  -  S)  (2) 

The  value  of  5  is  known,  for  its  tangent  is  -•    It  should  be  observed 

that  there  is  no  restriction  on  the  value  oi  6.  As  the  point  P 
moves  on  the  circle  OC,  the  circumference  is  twice  described  as  d 
varies  from  0°  to  360°,  but  the  diagram  for  other  positions  of  the 
point  P  is  in  no  case  essentially  different  from  Fig.  126. 

The  above  reasoning  and  the  diagram  involve  the  restriction 
_  that  both  o  and  6  are  positive  numbers.    While  it  is  possible  to 
supplement  the  reasoning  to  cover  the  cases  in  which  this  restric- 
tion is  removed,  it  is  unnecessary  as  the  analytical  proof  of  §74 
is  applicable  for  all  values  of  a  and  b. 

The  equation  of  the  circle  OC  in  any  position,  that  is,  for 
any  value  of  a  and  6,  positive  or  negative,  may  also  be  written 

in  the  form  

p  =  Va^  +  b^sm{d  +  i)  '      (3) 

in  which  e  is  the  angle  BOC  in  Fig.  126.    The  equation  of  the 

20 


306         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§160 

circle  OC  has  therefore  been  written  in  three  different  forms, 
namely  equations  (1),  (2)  and  (3)  above. 

Illustration  1.  From  the  above  we  know  that  the  equation 
p  =  6  cos  9  +  8  sin  9  represents  a  circle.  The  diameter  of  the 
circle  is  Va*  +  ft"  =  VSM"^  =  lOi  so  that  the  equation  of  the 
circle  may  also  be  written   p  =  10  cos  (9  —  i),  where  5  is  the  angle 

whose  tangent  is  -  =  ^  =   1.33.     From  a  table  of  tangents  S  =  53°  8', 

so  that  the  equation  of  the  circle  may  be  written  p  =  10  cos  (9  —  5l3  °8'). 

Illustration  2.  Write  the  equation  of  the  circle  p  =  cos  9  — 
-y/S  sin  9  in  the  form  p  =  c  cos  (9  —  S)  and  in  the  form  p  = 
c  sin  (9  +  e). 

Here  a  =  1,  6  =  —  -y/S,  c  =  \/a*  +  6«  =  2.  Hence  C  must  be  the 
point  (1,  —  \/3)  in  the  second  quadrant.  Then  5  =  angle  of  second 
quadrant  whose  tangent  is  ( —  -s/S/l),  or  120°.  Also  «  =  —  30°.  Hence 
the  required  equations  are  p  =  2  cos  (9  —  120°)  and  p  =  2  sin  (9  —  30°). 

The  result  of  this  section  should  also  be  interpreted  when  the  vari- 
ables are  x  and  y  in  rectangular  coordinates,  and  not  p  and  9  of 
polar  coordinates.  Thus,  y  =  a  cos  a;  is  a  sinusoid  with  highest  point 
or  crest  at  a;  =0,  2t,  iar,  .    .    .     Likewise,  y  =  b  sin  s  is  a  sinusoid 

with  crest  at  a;  =  y  -«-'  -~-'  .    .         The  above  demonstration  shows 

that  the  curve 

y  =  a  cos  X  +  b  sin  X 

is  identical  with  the  sinusoid 

y  =  Va'  +  6"  cos  (a;  -  hi)  =  \/a'-  +  ¥  sin  (a;  +  h) 

of  amplitude  \/a^  -\-  b-  and  with  the  crest  located  at  a;  =  hi,  or  at 

s  —  ^2,  where  hi  is,  in  radians,  the  angle  whose  tangent  is  ->  and  hi 

is,  in  radians,  the  angle  whose  tangent  is  r- 


Exercises 

1.  Put  the  equation  p  =  2  cos  9  +  2v'3  sin  9  in  the  form   p  = 
c  cos  (9  —  8)  and  find  the  value  of  h. 

2.  Put   the  equation  p  =  4  cos  9  +  \\/Z  sin  9  in  the  form  p  = 
ccos  (9  —  5). 

3.  Put  the  equation  p  =  —  4  cos  §  —  4  sin  9  in  the  form  p  - 
c  sin  (9  +  e), 


§161]  TRIGONOMETRIC  EQUATIONS  307 

4.  Put  the  equation  p  =  2-\/3  cos  6  +  2  sin  fl  in  the  form  p  = 
c  cos  (9  —  8). 

6.  Put  the  equation  p  =  3  cos  9  +  4  sin  9  in  the  form  p  = 
csin  (9  +  e).  Put  the  same  equation  in  the  form  p  =  c  cos  (9  —  'S). 
(S  is  the  angle  AOC,  Fig.  126.) 

6.  Put  the  equation  p  =  5  cos  9  +  12  sin  9  in  the  form  p  = 
o  sin  (9  +  e);  also  in  the  form  p  =  c  cos  (9  —  S). 

7.  Put  the  equation  (x  —  1)^  +  (y  —  ly  =  2  in.  the  form  p  = 
c  sin  (9  +  a)  and  determine  c  and  a. 

8.  Put  the  equation  (a;  +  1)^  +  (2/  —  \/3)'  =  4  in  the  form  p  = 
c  sin  (9  —  a)  and  determine  c  and  a. 

9.  Put  the  equation  (x  +  1)^  +  (y  +  Vs)^  =  4  in  the  form  p  = 
c  sin  (9  —  a)  and  determine  c  and  a. 

10.  Find  the  maximum  value  of  cos  9  —  \/3  sin  9,  and  determine 
the  value  of  9  for  which  the  expression  is  a  maximum. 

Suggestion  :  Call  the  expression  p.  The  maximum  value  of  p  is  the 
diameter  of  the  circle  p  =  cos  9  —  -y/s  sin  9.  The  direction  angle  of 
the  diameter  is  the  value  of  a  when  the  equation  is  put  in  the  form 
p  =  c  cos  (9  —  a). 

11.  Find  the  value  of  9  that  renders  p  =  f-v/S  cos  9  —  ^  sin  9  a 
maximum  and  determine  the  maximum  value  of  p. 

12.  Find  the  maximum  value  of  3  cos  t  +  isint. 

161.  Addition  Fonnulas  for  the  Sine  and  Cosine.  From  the 
preceding  section,  equations  (1),  (2)  and  (3),  we  know  that  the 
equation  of  the  circle  OC,  Fig.  127,  may  be  written  in  any  one  of 
the  forms 

p  =  a  cos  6  +  6  sin  0,  (1) 

p  =  c  sin(,e-  e),  (2) 

p  =  c  cos  (e-5).  (3) 

Hence,  for  all  values  oi  9,  d,  and  e, 

sin  (6  —  e)  =  -  cos  5  +  -  sin  6,  (4) 

cos  (9  -  5)  =  -cose  +  - sin  9,  (6) 

In  each  of  these  equations  c  =  -^/a"  +  h^.  The  letters  a  and  6 
stand  for  the  co6rdinates  of  C  irrespective  of  their  signs  or  of 
the  position  of  C, 


308        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§161 

Since  (4)  and  (5)  are  true  for  all  values  of  6,  they  are  true  when 
»  =  0°  and  when  0  =  90°. 

First,  Let  0  =  0°  in  (4). 

Then  a/c  =  sin  (—  e)  =  —  sin  e  by  §60,  (6) 

-Second,  let  B  =  90°  in  (4), 

Then  6/c  =  sin  (90°  -  e)  =  cos  «  (7) 

Substituting  (6)  and  (7)  in  (4)  we  have 

sin  (5  —  e)  =  sin  0  cos  e  —  cos  6  sin  e  (8) 


Fig.  127. — The  circle  p  =  c  cos  (9  —  5)  or  p  =  sin  c  (fl  —  e)  used 
in  the  proof  of  the  addition  formulas.  Note  that  e  =  90°  +  « 
which  is  also  true  for  negative  angles,  namely   Si  =  90°  +  ei. 

In  like  manner  upon  letting  0  =  0  and  0  =  90°  in  succession  in 
(5)  we  have 

-  =  cos  (-  5)  =  cos  5,  by  §60.  (9) 


=  cos  (90  —  5)  =  sin  5. 


Substituting  (9)  and  (10)  in  (5)  we  obtain 

cos  {8  —  B)  =  cos  9  cos  6  +  sin  0  sin  5 


(10) 


(11) 


§162]  TRIGONOMETRIC  EQUATIONS  309 

Since  these  are, true  for  all  values  of  S  and  e,  put  5  =  (-^Si)  and 
e  =  (  —  ei).    Then  by  §60,  these  equations  become 

sin  (6  +  ei)  =  sin  0  cos  ei  +  cos  6  sin  €i  (12) 

cos  (0  +  Si)  =  cos  6  cos  5i  —  sin  9  sin  5i  (13) 

To  aid  in  committing  these  four  important  formulas  (8),  (11), 

(12)  and  (13)  to  memory,  it  is  best  to  designate  in  each  case  the 

angles  by  a  and  |3,  and  write  (12)  and  (13)  in  the  form 

sin  (a  +  /3)  =  sin  o:  cos  /3  +  cos  a  sin  /3  (14) 

cos  (a  +  /3)  =  cos  a  cos  |8  —  sin  a  sin  /3  (15) 

and  also  write  (8)  and  (11)  in  the  form 

sin  (a  —  P)  =  sia  a  cos  /3  —  cos  a  sin  /3  (16) 

cos  (a  —  |3)  =  cos  a  cos  /3  +  sin  q:  sin  j3  (17) 

The  four  formulas  (14),  (15),  (16)  and  (17)  must  be  committed  to 
memory.  They  are  called  the  addition  fonnulas  for  the  sine  and 
cosine.  The  above  demonstration  shows  that  the  addition  for- 
mulas are  true  for  all  values  of  a  and  fi. 

By  the  above  formulas  it  is  possible  to  compute  the  sine  and 
cosine  of  75°  and  15°  from  the  following  data: 

sin  30°  =  i  sin  45°  =  iV2 

cos  30°  =  i  V3  cos  45°  =  iV2 

Thus 
sin  75°  =  sin  (30°  +  45°)  =  sin  30°  cos  45°  +  cos  30°  sin  45° 

=  HV2  +  |\/3iV2  =  iV^(V'3  +  1) 
Likewise 

sin  15°  =  sin  (45°  -  30°)  =  i-s/2(\/3  -  1) 

162.  Addition  Formula  for  the  Tangent.    Dividing  the  mem- 
bers of  (14)  §161  by  the  members  of  (15)  we  obtain 

,     ,   a\       sin  (a  -t-  /3)      sin  a  cos  |8  -h  cos  a  sin  |3      ,, , 

tan  (a  +  p)  = -, — —37-  = 5 ; ; — 5      ^1; 

cos  (o -t- p)       cos  acosp  —  sm  asmp 

Dividing  numerator  and  denominator  of  the  last  fraction   by 

cos  a  cos  /3 

sin  a  cos  /3      cos  a  cos  |8 

tan  ia  +  ^)  =  ^^E^^l_Jo[^^  (2) 

cos  a  cos  fi  _  sin  a  sin  fi 

cos  a  cos  ^      cos  acosfi 


310        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§163 

or 

,     ,    ..        tan  a  +  tan  /3  ,„, 

tan  (a  +  B)  = — ^  (3) 

y     I  t^'      I  -  tan<»tan|3 

Likewise  it  can  be  shown  from  (16)  and  (17),  §161,  that 

.       ,         „        tana  — tan  ^  ,  . 

tan  (a  -  |3)  =  — t— r— ^  (4) 

^  '       I  +  tanatan|3 

Equations  (3)  and  (4)  are  the  addition  formulas  for  the  tangent. 

Exercises 

1.  Compute  cos  75°  and  cos  15°. 

2.  Compute  tan  75°  and  tan  15°. 

3.  Write  in  simple  form  the  equation  of  the  circle 

p  =  sin  6  +  cos  B. 

4.  Put  the  equation  of  the  circle  p  =  3  sin  9  +  4  cos  6  in  the  form 
p  =  c  sin  (9  +  9i)  and  find,  from  the  tables  or  by  the  slide  rule,  the 
value  of  ©i. 

6.  Derive  a  formula  for  cot  (a  +  p). 

6.  Prove  cos  (s  +  t)  cos  {s  —  t)  =  cos''  s  —  sin^  t. 

7.  Express  in  the  form  c  cos  (a  —  b)  the  binomial  3  cos  a  +  4  sin  o. 

8.  Express  in  the  form  c  sin  (a  +  6)  the  binomial  5  cqs  o  +  12  sin  a. 

9.  Find  the  coordinates  of  the  maximum  point  or  crest  of  the  sinus- 
oid y  =  sin  X  +  -\/3  cos  x.  [First  reduce  the  equation  to  the  form 
2/  =  c  sin  (a;  +  a)]. 

163.  Functions  of  Composite  Angles.  The  sine,  cosine,  or 
tangent  of  the  angles  (90°  —  d),  (90°  +  6),  (180°  -  6),  (180°  +  6), 
(270°  —  6),  (270°  +  d)  can  be  expressed  in  terms  of  functions 
of  6  alone  by  means  of  the  addition  formulas  of  §§161  and  162. 
Thus,  write    . 

sia  (a  +  /?)  =  sin  a  cos  /3  +  cos  a  sin  /3  (1) 

cos  {a  +  fi)  =  cos  a  cos  ;8  —  sin  a  sin  j3  (2) 

Put  a  =  180°,  and  /3  =  +  0;  then  (1)  and  (2)  become,  re- 
spectively, 

sin  (180°  ±  0)  =  T  sin  e  (3) 

cos  (180°  ±6)  =  ~  Gosd  (4) 


TRIGONOMETRIC  EQUATIONS 


311 


Also  in  (1)  and  (2)  put  a  =  90°,  and  P  =  ±6,  then  (1)  and  (2) 
become,  respectively, 

sin  (90°  ±  6)  =       cos  6  (5) 

cos  (90°  ±  6)  =  +  sine  ■                  (6) 

By  division  of  (3)  by  (4)  and  of  (5)  by  (6), 

tan  (180°  ±6)  =  +  tan  6»,  (7) 

tan  (90°  ±  6)  =  +  cot  d.  (8) 

In  a  similar  manner  all  of  the  results  given  in  the  following  table 
may  be  proved  to  be  true. 


(-ft./l)  Pa 


AC  ft,  A) 


(A. ft) 


P(h.k) 


Pt(.-h,-h) 


P,(h.-k) 


(-ft.-ft)    Pa 


P,  (k.-h) 


A  B 

Fig.  128. — An  angle   9  combined  with  an  even  number  of  right 
angles,  (A)  and  wijh  an  odd  number  of  right  angles,  (B) . 

TABLE  VII 

Functions  of  6  Coupled  with  an  Eeen  or  with  an  Odd  Number  of 
Bight  Angles 


-  e 

90° -9 

90°+  9 

180°-  9 

180°+  9 

270°-  9 

270°+  8 

sin 

—  sin  9 

cos  9 

cos  9 

sin  9 

—  sin  9 

—  cos  9 

—  cos  B 

cos 

'    cos  e 

sin  9 

—  sin  9 

—  cos  9 

—  cos  9 

—  sin  B 

sin  8 

tan 

—  tan  e 

cote 

—  cot  e 

—  tan  9 

tan  9 

cot  9 

-  cot  9 

AU  of  the  above  results  can  be  included  in  two  simple  state- 
ments.   For  this  purpose  it  is  convenient  to  separate  into  different, 


312         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§163 

classes  the  composite  angles  that  are  made  by  coupling  0  with 
an  odd  number  of  right  angles,  as  (90°  +  fl),  (ff  -  90°),  (270°  -  6), 
(450°  +  6),  etc.,  and  those  composite  angles  that  are  made  by 
coupling  6  with  an  even  number  of  right  angles,  as  (180°  +  6) 
(180°  -  6),  (360°  -  6),  (-  6),  etc.  Note  that  0  is  an  even 
number,  so  that  ( —  9)  or  (0°  —  6)  falls  into  the  second  class  of 
composite  angles.    We  can  then  make  the  following  statements : 

Theorems  on  Functions  of  Composite  Angles 

Think  of  the  original  angle  6  as  an  angle  of  the  first  quadrant: 

I.  Any  function  of  a  composite  angle  made  by  coupling  B  {by 
addition  or  subtraction)  with  an  even  number  of  right  angles,  is 
equal  to  the  same  function  of  the  original  angle  6,  with  an  algebraic 
sign  the  same  as  the  sign  of  the  function  of  the  composite  angle  in 
its  quadrant. 

II.  Any  function  of  a  composite  angle  made  by  coupling  0  (by 
addition  or  subtraction)  with  an  odd  number  of  right  angles,  is  eqital 
to  the  co-function  of  the  original  angle  B,  with  an  algebraic  sign 
the  same  as  the  sign  of  the  function  of  the  composite  angle  in  its 
quadrant. 

For  example,  let  the  original  angle  be  6,  and  the  composite  angle  be 
(180°  +  8).  Take  any  function  of  (180°  +  $),  say  tan  (180°  +  6),  it  is 
equal  to  +  tan  6,  the  sign  +  being  the  sign  of  the  tangent  in  the  quad- 
rant of  the  composite  angle  (180°  +  8),  or  third  quadrant.  Likewise 
cot  (270°  +  6)  must  equal  the  negative  co-function  of  the  original 
angle,  or  —  tan  0,  the  algebraic  sign  being  the  sign  of  the  cotangent  in 
the  quadrant  of  the  composite  angle  (270°  -|-  9),  or  fourth  quadrant. 
In  the  above  statements  it  has  been  assumed  that  the  angle  fl  is  an 
angle  of  the  first  quadrant.  This  is  merely  for  the  convenience  of 
determining  signs,  for  the  results  stated  in  itaUcs  are  true,  no  matter 
in  what  quadrant  9  may  actually,  he. 

Exercise 

Given  sin  30°  =  J,  cos  30°  =  JVS,  tan  30°  =  iVS,  cot  30°  =  \/3, 
find  the  sine,  cosine,  and  tangent  of  each  of  the  following  angles  by 
means  of  the  above  Theorems  on  Functions  of  Composite  Angles: 
(a)  150°;  ^b)  210°;  (c)  240°;  (d)  300°;  (e)  330°;  (/)  120°;  (g)  60°; 
(h)  -30°. 


§164] 


TRIGONOMETRIC  EQUATIONS 


313 


164.  Angle  that  a  Given  Line  Makes  with  Another  Line.    The 

slope  m  of  the  straight  line  y  =  mx  +  b  is  the  tangent  of  the 
direction  angle,  that  is,  the  tangent  of  the  angle  that  the  line  makes 
with  OX.  If  Li  and  L^  are  any  two  lines  in  the  plane,  the  angle 
that  Li  makes  with  Lj  is  the  positive  angle  through  which  L^  must 
he  rotated  about  their  point  of  intersection  in  order  that  Li  may 
coincide  with  Li.  Represent  the  direction  angles  of  two  straight 
lines 

y  =  miX  +  bi  (1) 

y  =  mix.  +  hi  (2) 

by  the  symbols  di  and  6^.    Then,  through  the  intersection  of  the 

lines  pass  a  line  parallel  to  the  OX-axis,  as  shown  in  Fig.  129. 

Call  0  the  angle  that  the  line  Li  makes  with  La)  that  is,  the  positive 


^ 

'\> 

'> 

A 

0 

V 

\- 

\ 

\L,. 

I      Fig.  129. — The  angle  <l>  that  a  line  Li  makes  with  -La.      J 

angle  through  which  La,  considered  as  the  initial  line,  must  be 
turned  to  coincide  with  the  terminal  position  given  by  Li.  If 
9i  >  Bi,  then  4>  =  Bi-  6^,  but  if  fla  >  Oi,  then  0  =  180°  -  (Sj  -  di). 
In  either  case  (by  equations  (7),  §163,  and  (3),  §60 

tan  <i>  =  tan  {di  -  d^).  (3) 

That  is, 

tan  di  —  tan  82  ,,. 

(4) 


or 


tan  (b  = 


tan  <j>  = 


1  +  tan  61  tan  62' 
nil  —  nia 


(5) 


I  +  niinia 

The  condition  that  the  given  lines  (1)  and  (2)  are  parallel  is 
obviously  that 

mi  =  ma  (6) 

Thus,  the  lines  y  =  5x  +  7  and  y  =  5x  —  11  ar«  parallel. 


314        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§164 

The  condition  that  the  given  lines  (1)  and  (2)  are  perpendicular 
to  each  other  is  that  tan  4>  shall  become  infinite;  that  is,  that  the 
denominator  of  (5)  shall  vanish.  Hence  the  condition  of  perpen- 
dicularity is 

1  +  miW2  =  0, 


m:  =  -  ^-  (7) 

Therefore,  in  order  that  two  lines  may  be  perpendicular  to  each 
other,  the  slope  of  one  line  must  be  the  negative  reciprocal  of  the  slope 
of  the  other  line. 

Thus  the  lines  y  =  %x  —  A  and  y  =  —  fa;  +  2  are  per- 
pendicular. 

Exercises 

1.  Find  the  tangent  of  the  angle  that  the  first  line  makes  with 
the  second  line  of  each  set: 

[a)  y  =  2x  +  Z,  y  =  x  +  2, 

{h)y  =  Zx  -Z,  y  =  2x  +  1, 

(c)  y  =  ix  +  5,  y  =  Zx  -  1, 

id)  y  =  lOx  +  I,  y  =  Ux  -  1, 

2.  Find  the  angle  that  the  first  line  of  each  pair  makes  with  the 
second: 

(a)  y  =  X  +5,  y  =  -  a;  -|-  5. 

(6)  2/  =  Ja;  -H  6,  y  =   -  2x. 

(c)  2/  =  2a;  +  4,  y  =  x  +  1. 

(d)  2x+Zy  =  \,  Ix  =y  =  \. 

(e)  2i  4-  42/  =  3,  3a;  4-  62/  =  7. 
(/)  2x  +Ay  =  3,  6a;  -  32/  =  7. 

3.  Find  the  angle,  in  each  of  the  following  cases,  that  the  first 
line  makes  with  the  second:  - 

(o)  2/=  x/Vz  +4,  2/  =  V3x+  2. 

(6)  y  =  a;/\/3  -|-  1,  y  =  VZx-  4. 

(c)  y  =  y/Zx  -  6,  2/  =  s/Zx-  Z. 

4.  Find  the  angle  that  2i/  —  6a;  -1-  7  =  0  makes  with  y  +  2x  + 
7=0  and  also  the  angle  that  the  second  line  makes  with  the  first. 


§165]  TRIGONOMETRIC  EQUATIONS  315 

166.  The  Functions  of  the  Double  Angle.  The  addition 
formulas  for  the  sine,  cosine,  and  tangent  reduce  to  formulas  of 
great  importance  for  the  special  case  fi  =  a. 

Thus  sin  (a  +  a)  =  sin  a  cos  a  +  cos  a  sin  a, 

or  sin  2a  =  2  sin  a  cos  a.  (1) 

Also  cos  (a  +  a)  =  cos  a  cos  a  —  sin  a  sin  a 

which  can  be  written  in  the  three  forms: 

cos  2  a  =  cos^  a  —  sin^  a,  (2) 

cos  2  a  =  2  cos^  oi  —I,  (3) 

cos  2  o!  =  I  —  2'  sin^  a.  (4) 

Forms    (3)    and    (4)    are   obtained   from    (2)    by  substituting, 
respectively,  sin^  a  =  1  —  cos''  a  and  cos^  a  =  1  —  sin^  a. 
Equations  (3)  and  (4)  are  frequently  useful  in  the  forms : 

— ,  .       I  +  cos  2a;  ,g, 

(6) 
Again 


2 

• 

sin^ 

a 

= 

I  —  cos  2a 

2 

tan  {a  +  a) 

= 

tan  a  +  tan  a 
1  —  tana  tan  a 

fan 

2  tana 

■•""'""    I  -  tan»  a  ^^^ 

166.  The  Functions  of  the  Half  Angle.  From  (6)  and  (5)  of 
§165  we  obtain,  after  replacing  a  by  u/2  and  extracting  the 
square  root, 

sin  (u/2)  =    ±v'(i  —  cos  u)/2,  (1) 

cos  (u/2)  =    ±  \/(i  +  cos  u)  /2 .  (2) 

Dividing  (1)  by  (2),  we  obtain 

*      /-    /  ^       j_  1  /i  -  COS  u       j^  I  -  cos  u       ^      sin  u     .    ,„. 

tan  (u/2)  =  ±  V — ; =  ± ^^ =  ±  — i (3) 

^  '   '  » I  +  cos  u  sm  u  I  +  cos  u 

Formulas  (1),  (2)  and  (3)  have  many  important  applications  in 
mathematics.    As  a  simple  example,  note  that  the  functions  of  15° 


316        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§167 

may  be  computed  when  the  functions  of  30°  are  known.    Thus 

cos  30°  =  (1/2)  VS 
therefore     sin  15°  =  \/(l  -  cos  30°)/2  =  V'l/2  -  (1/4)^3- 
Also  cos  15°  =  Vl/2  + (1/4)^3. 

Likewise  by  (3) 

tan  15°  =  L^^#^  =  2  -  V3. 

Exercises 

1.  Compute  sin  60°  from  the  sine  and  cosine  of  30°. 

2.  Compute  sine,  cosine,  and  tangent  of  221°. 

3.  If  sin  X  =  2/5,  find  the  numerical  value  of  sin  2x,  cos  2x,  and 
tan  2x. 

4.  Show  by  expanding  sin  (x  +  2x)  that  sin  3a;  =  3  sin  a;  —  4  sin  'x. 

,    _,         .       „        3  tan  x  —  tan'  x 

0.  Prove  tan  3a;  =  — 5 5-: — ; 

1—3  tan*  X 

6.  Show  that  sin  29/sin  e  —  cos  29/cos  9  =  sec  8. 

7.  Show  that 


Ism 2  +  cos^)     =  1  +  sin  e. 


8.  Show  that  cos  29(1  +  tan  29  tan  B)  =  1. 

9.  If  sin  A  =  3/5,  calculate  sin  {A/2). 

10.  Prove  that  tan  (7r/4  +  9)  =  ^  _  ^  g- 

11.  Prove  that  tan  (ir/4  -  9)'  =  (1  -  tan  9)/(l  +  tan  9). 

12.  Show  that  sec  9  +  tan  9  = ^^— • 

cos  9 

. »    p.,        . ,    .  1  +  2  sin  a  cos  a      cos  a  +  sin  a 

13.  Show  that  -■    x—- ;-= —  = ■ — -. 

cos*  a  —  sm*  a        cos  a  —  sm  o 

14.  Show  that  sec  9  +  tan  9  =  tan 


[i+g 


16.  Show  that  — ^—j — j :r-^  —  tan  A  tan  B. 

cot  A  +  cot  B 

16.  Prove  that  cos  (s  +  t)  cos  {s  —  t)  +  sin  (s  +  t)  sin  (s  —  i)  = 

cos  2t. 

167.  Sums  and  Differences  of  Sines  and  of  Cosines  Expressed 
as  Products.  The  following  formulas,  which  permit  the  substi- 
tution of  a  product  for  a  sum  of  two  sines  or  of  two  cosines,  are 


§167]  TRIGONOMETRIC  EQUATIONS  317 

important  in  many  transformations  in  mathematics,  especially  in 
the  calculus.  They  are  immediately  derivable  from  the  addition 
formulas.  Thus,  by  the  addition  formulas  (14)  and  (16),  §161,  we 
obtain 

sin  (a  +  6)  +  sin  (a  —  6)  =  2  sin  o  cos  6. 

Likewise  by  subtraction  of  the  same  formulas 

sin  (a  +  6)  —  sin  (a  —  b)  =  2  cos  a  sin  b. 

By  the  addition  and  subtraction,  respectively,  of  the  addition 
formulas  for  the  cosine  there  results 

cos  (a  +  6)  +  cos  (a  —  b)  =  2  cos  a  cos  6. 

cos  (a  +  6)  —  cos  {a  —  b)  =  —  2  sin  a  sin  6. 
These  formulas  can  be  written 

sin  o  cos  b  —       5  [sin  (o  +  6)  +  sin  (a  —  6)],         (1) 
cos  a  sin  6  =       |  [sin  (a  +  6)  —  sin  (o  —  b)],  (2) 

cos  a  cos  6  =       2  [cos  (a  +  b)  +  cos  (a  —  6)],         (3) 
sin  a  sin  6  =  —  5  [cos  (a  +  6)  —  cos  (a  —  6)].  (4) 

Represent  (a  +  6)  by  a  and  (a  —  b)  by  /?. 
Then  o  =  (a  +  /3)  /2  and  b  =  (a  -  j8)/2 
Hence  the  above  formulas  become 

sin  a  +  sin  j3  =       2  sm cos  -'  (5) 

•     n  a  +  0     .     a  —  0  ,„, 

sm  a  —  sin  fl  =       2  cos sm  >  (6) 

22 

cos  a  +  cos  /3  =       2  cos cos  >  (7) 

2  2 

cos  a  —  cos  /3  =  —  2  sm sm  ^-  (8) 

2  2 

The  principal  use  of  these  formulas  is  in  certain  transformations 
in  the  Calculus.  A  minor  use  is  in  adapting  certain  formulas  to 
logarithmic  work  by  replacing  sums  and  differences  by  products. 

These  formulas  should  not  be  committed  to  memory.  They 
can  be  derived  in  a  moment  when  needed  by  recalling  their 


318        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§168 

connection  with  the  addition  formulas.    Formula  (2)  is  really- 
contained  in  formula  (1).    For  by  (1) 

cos  a  sin  6  =  sin  6  cos  a 

=  5  [sin(&  +  a)  +  sin  (6  —  a)] 
=  5  [sin  (a  +  6)  —  sin  (a  —  6)], 
since  sin(—  B)  =  —  sin  & 

Exercises 

Express  as  the  sum  or  difference  of  sines  or  cosines: 

1.  sin  5x  cos  2x.  6.  sin  3x  sin  7x. 

2.  cos  3a;  sin  7x.  7.  cos  3a;  cos  8x. 

3.  cos  4a;  cos  x.  8.  cos  5a;  sin  2x. 

4.  sin  5x  sin  2x.  9.  sin  3a;  cos  lOx. 
6.  sin  3x  cos  5x.  10.  cos  2x  cos  6x. 

168.*  Graph  of  y  =  sin  2x,  y  =  sin  nx,  etc.  Since  the  substi- 
tution of  nx  for  X  in  any  equation  multiplies  the  abscissas  of  the 
curve  by  1/n,  or  («>!)  shortens,  or  contracts,  the  abscissas  of  all 
points  of  the  curve  in  the  uniform  ratio  n  :  1,  the  curve  y  =  sin  2x 
must  have  twice  as  many  crests,  nodes,  and  troughs  in  a  given 
interval  of  x  as  the  sinusoid  y  =  sin  x.  The  curve  y  =  sin  2x  is 
therefore  readily  drawn  from  Fig.  59  as  follows:  Divide  the  axis 
OX  into  twice  as  many  equal  intervals  as  shown  in  Fig.  59  and 
draw  vertical  lines  through  the  points  of  division.  Then  in  the 
new  diagram  there  are  twice  as  many  small  rectangles  as  in  the 
original.  Starting  at  0  and  sketching  the  diagonals  (curved  to 
iit  the  alignment  of  the  points)  of  successive  cornering  rectangles, 
the  curve  y  =  sin  2x  is  constructed.  It  is,  of  course,  the  ortho- 
graphic projection  of  J/  =  sin  x  upon  a  plane  passing  through 
the  F-axis  and  making  an  angle  of  60°  (the  angle  whose  cosine  is 
1/2)  with  the  x2/-plane.  The  curve  y  =  cos  2x  is  sunilarly  con- 
structed. In  each  of  these  cases  we  see  that  the  period  of  the 
function  is  t  and  not  2ir. 

169.*  Graph  of  p  =  sia20,p  —  cos  20,  etc.  The  curve  p  =  cos  6 
is  the  circle  of  diameter  unity  coinciding  in  direction  with  the  axis 
OX.  We  have  already  emphasized  that  as  d  varies  from  0°  to 
360°  the  circk  is  twice  drawn,  so  that  the  curve  consists  of  two 


§170] 


TRIGONOMETRIC  EQUATIONS 


319 


superimposed  circular  loops.  Now  p  =  cos  2d  wiU  be  found  to 
consist  of  four  loops,  somewhat  analogous  to  the  leaves  of  a  four- 
leafed  clover,  but  each  loop  is  described  but  once  as  6  varies  from 
0°  to  360°.  The  curve  p  =  cos  36  is  a  three-looped  curve,  but  each 
loop  is  twice  drawn  as  S  varies  from  0°  to  360°.  Also  p  =  cos  116 
has  eleven  loops,  each  twice  drawn,  while  p  =  cos  126  has  twenty- 
four  loops,  each  one  described  but  once,  as  6  varies  from  0°  to  360°. 
The  curves  p  =  cos  2  6,  p  =  sin  39,  p  =  sin  6/2  should  be  drawn 
by  the  student  upon  polar  coordinate  paper. 

170.*  Graph  of  y  =  sln^x,  and  y  =  cos^x.    The  graphs  y  =  sin'  x 
and  y  =  cos''  x  have  important  applications  in  science.     The  following 


E 


Y 
A 

s 

P^-^? 

■'^.^-L 

— 

— 

— 

— 

— 

ci._4V. 

\ 

/ 

N 

rr-l     M 

N^ 

/ 

N 

ff|--|    - 

/ 

s 

/    "^  \    \3^ 

1      1   / 

S^ 

/ 

"—    ^^"^-^    \ 

/ 

~~—-^^^~^ 

<. 

f^ 

^"^^^r 

0 

y 

/ 

/ 

H 

Fig.  130. — The  graph  oi  y  =  cos^  x. 

graphical  method  offers  an  easy  way  of  constructing  the  curves  and  it 
illustrates  a  number  of  important  properties  of  the  functions  involved. 
We  shall  first  construct  the  curve  y  =  cos'  x.  At  the  left  of  a  sheet  of 
8  J  X  11-inch  paper  draw  a  circle  of  radius  36/57r  ( =  2.30)  inches,  (OA, 
Fig.  130).  Lay  off  the  angles  9  from  OA,  as  initial  line,  correspond- 
ing to  equal  intervals  (10°)  of  the  quadrant  APE  as  shown  in  the 
figure.  Let  the  point  P  mark  any  one  of  these  equal  intervals. 
Then  dropping  the  perpendicular  AB  from  A  upon  OP,  the  dis- 
tance OB  is  the  cosine  of  6,  if  OA  be  called  unity.  Dropping  a 
perpendicular  from  B  upon  OA,  the  distance  OC  is  cut  off,  which  is 
equal  to  OB'  or  cos'  e,  since  in  the  right  triangle  OB  A,  OB'  —  OCOA 
=  OC-1.  Making  similar  constructions  for  various  values  of  the  angle 
e,  say  for  every  10°  interval  of  the  arc  APE,  the  hne  OA  is  divided  at 
a  number  of  points  proportionally  to  cos'  e.     Draw  horizontal  lines 


320         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§171 

through  each  point  of  division  of  OA.  Next  divide  the  axis  OX  into 
intervals  equal  to  the  intervals  of  8  laid  off  on  the  arc  APE.  Since  the 
radius  of  the  circle  OA  was  taken  to  be  (SB/Sir)  inches,  an  interval  of 
10°  corresponds  to  an  arc  of  length  2/5  inch,  which  therefore  Inust 
be  the  length  of  the  equal  intervals  laid  off  on  OX.  Through  each  of 
the  points  of  division  of  OX  draw  vertical  lines,  thus  dividing  the 
plane  into  a  large  number  of  small  rectangles.  Starting  at  A  and 
sketching  the  diagonals  of  successive  cornering  rectangles,  the  locus 
ARS  oi  y  =  cos'  x  is  constructed. 

From  Fig.  130,  it  is  seen  that  B  always  lies  at  the  vertex  of  a  right- 
angled  triangle  of  hypotenuse  OA.  Thus  as  P  describes  the  circle  of 
radius  OA,  B  describes  a  circle  of  radius  OA/2.  Therefore  the  curve 
ABS  is  related  to  the  small  circle  ABO  in  the  same  manner  that 
the  curve  of  Fig.  59  is  related  to  its  circle;  consequently  the  curve 
ARS  of  Fig:  130  is  a  sinusoid  tangent  to  the  X-axis.  Thus  the  graph 
y  =  cos'  a;  is  a  cosine  curve  of  amplitude  1/2  and  wave  length  or  period 
IT,  lying  above  the  X-axis  and  tangent  to  it. 


B.  PLANE  TRIANGLES:  CONDITIONAL  EQUATIONS 

171.  Law  of  Sines.     The  first  of  the  conditional  equations  per- 
taining to  the  oblique  triangle  is  a  proportion  connecting  the  sines 


Fig.  131. — Derivation  of  the  law  of  sines  and  the  law  of  cosines. 


of  the  three  angles  of  the  triangle  with  the  lengths  of  the  respec- 
tive sides  lying  opposite.  Call  the  angles  of  the  triangle  A,  B,  C, 
and  indicate  the  opposite  sides  by  the  small  letters  a,  b,  c,  respec- 
tively. From  the  vertex  of  any  angle,  drop  a  perpendicular  p 
upon  the  opposite  side,  meeting  the  latter  (produced  if  necessary) 


§172]  TRIGONOMETRIC  EQUATIONS  321 

at  D.    Then,  by  the  properties  of  right  triangles,  we  have,  in 
either  Fig.  131  (1)  or  131  (2), 

p  =  c  sin  DAB.  (1) 

From     A  BDC, 

p  =  a  sin  C.  (2) 

But, 

sin  DAB  =  sin  A,  Fig.  131  (1) 

=  sin  (180°  -  A),  Fig.  131  (2) 
=  sin  A. 
Therefore  p  =  c  sin  A  =  » sin  C,  (3) 

or  a/sin  A  =  c/sin  C.  (4) 

In  like  manner,  by  dropping  a  perpendicular  from  A  upon  a,  we 
can  prove 

b/sin  B  =  c/sin  C.  (5) 

Therefore  a/sin  A  =  b/sin  B  =  c/sin  C  (6) 

Stated  in  words,  the  formula  says:  In  any  oblique  triangle  the 
sides  are  proportional  to  the  sines  of  the  opposite  angles. 

Geometrically:  Calling  each  of  the  ratios  in  (6)  2B,  it  is  seen 
from  Fig.  131  (2)  that  R  is  the  radius  of  the  circumscribed  circle 
since  c/  sin  C  =  2R  can  be  deduced  from  the  triangle  BAE.  Similar 
construction  can  be  made  for  the  angle  B  or  A. 

172.  Law  of  Cosines.  From  plane  geometry  we  have  the  theo- 
rem: The  square  of  any  side  opposite  an  acute  angle  of  an  oblique 
triangle  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides  di- 
minished by  twice  the  product  of  one  of  those  sides  by  the  projection 
of  the  other  side  on  it.    Thus,  in  Fig.  131  (1), 

o2  =  6'2  +  c^  -  2bd.  (1) 

But  -      d  =  c  cos  A. 

Therefore  a^  =  b^  +  c'^  -  2bc  cos  A,  (2) 

Likewise  we  learn  from  geometry  that  the  square  of  any  side  oppo- 
site an  obtuse  angle  of  an  oblique  triangle  is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides  increased  by  twice  the  product  of  one  of 

21 


322         ELEMENTARY  MATHEMATICAL  ANALYSIS      (§172 

those  sides  by  the  projection  of  the  other  on  it.     Thus,  in  Fig.  131  (2) , 
a2  =  62  +  c2  +  2bd  (3)  ' 

But  d  =  c  cos  DAB  =  c  cos  (180  —  A)  =  —  c  cos  A. 

Therefore  (3)  becomes 

a2  =  b''  +  c^  -  2bc  cos  A.  (4) 

This  is  the  same  as  (2),  so  that  the  trigonometric  form  of  the  geo- 
metrical theorem  is  the  same  whether  the  side  first  named  is  oppo- 
site an  acute  or  opposite  an  obtuse  angle. 
In  the  same  way  we  may  show  that,  in  any^  triangle 

b2=  c2-|-a2 -2cacosB,  (5) 

c2  =  a^-l-b^  -  2ab  cosC.  (6) 

Independently  of  the  theorem  from  plane  geometry,  we  note 
from  Fig.  131  (1) 

a^  =  (b  -  dy  -f-  p2  =  (6  _  dy  +  c''  -  d^ 

=  62  +  (.2  _  -2,bd 

=  fe''  -h  c^  -  26c  cos  A. 
From  131  (2) 

o"  =  (6  -I-  dy  +  p2  =  (6  -h  dy  -I-  c2  -  d^ 
=  6='  -I-  c2  -I-  2bd 
=  62  +  0^-1-  26c  cos  DAB 
=  62-)-c'  -  26c  cos  A, 
since  DAS  =  180°  -  A  and  cos  (180°  -  A)  =  -  cos  A. 

Second   Phoop:    Since   any  side  of  an  oblique  triangle  is 
the  sum  of  the  projections  of  the  other  two  sides  upon  it,  the 
angles  of  projection  being  the  angles  of  the  triangle,  we  have 
a  =  b  cos  C  -|-  c  cos  B, 
b  =  c  cos  A  -f-  a  cos  C,  (7) 

c  =  a  cos  B  -h  b  cos  A. 
Multiply  the  first  of  these  equations  by  a,  the  second  by  6, 
the  third  by  c,  and  subtract  the  second  and  third  from  the  first. 
The  result  is 

a^  —  b^  —  c^  —  ah  cos  C  -\-  ca  cos  B 

—  6c  cos  A  —  ab  cos  C 

—  ca  cos  B  —  be  cos  A 
=  —  26c  cos  A, 

or  a^  =  6^  -h  c''  —  26c  cos  A. 


§173] 


TRIGONOMETRIC  EQUATIONS 


323 


173.  Law  of  Tangents.  An  important  relation  results  if  we 
take  formula  (5)  §171  by  composition  and  division.  First 
write  the  law  of  sines  in  the  form 


sin  A 
sin  jB' 


(1) 


Then,  by  composition  and  division,  the  sum  of  the  first  anteced- 
ent and  consequent  is  to  their  difference  as  the  sum  of  the  second 
antecedent  and  consequent  is  to  their  difference;  that  is 

a  +  h  _  sin  A  +  sin  B,  ,„, 

a  —  &      sin  A  —  sin  B 

Expressing  the  sums  and  difference  on  the  right-hand  side  of  (2)  as 
products  by  means  of  the  formulas  (5)  and  (6)  of  §167,  we  obtain 

a  +  b  ^  2  sin  i(,A  +  B)  cos  i(A  -  £)', 
a-h      2  cos  K^  +  B)  sin  |(A  -  B) 

or  simplifying  and  replacing  the 
ratio  of  sine  to  cosine  by  the  tan- 
gent, we  obtain 


(3) 


a-l-b  ^  tan  \{k  +  B). 
a  -  b      tan  J(A  -  B) 

In  like  manner  it  follows  that 

b  4-  c     tan  |(B  4-  C). 
b  -  c      tan  KB  -  C) 

0  4-  a  _  tan_|(C_+A) 
0  —  a      tan  KC  —  A) 


(4) 


Fig.  132. — Geometrical 
(R\  derivation     of    law    of    tan- 
gents. 


Expressed  in  words:  In  any  triangle,  the  sum  of  two  sides  is  to 
their  difference,  as  the  tangent  of  half  the  sum  of  the  angles  opposite 
is  to  the  tangent  of  half  of -their  difference. 

Geometrical  Proof:  From  any  vertex  of  the  triangle  as  center, 
say  C,  draw  a  circle  of  radius  equal  to  the  shortest  of  the  two  sides  of 
the  triangle  meeting  at  C,  as  in  Fig.  132.  Let  the  circle  meet  the  side 
a&t  R  and  the  same  side  produced  at  E.  Draw  AE  and  AR.  Call 
the  angles  at  A,  a,  and  /3,  as  shown.  Then  BE  =  a  +  6  and 
BR  =  a  -  b.    Also 

a  +  P  =  A, 


324        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§174 

and  /.  CRA  =  0  +  B  (the  external  angle  of  a  triangle  RAB  is  equal  to 
the  sum  of  the  two  opposite  interior  angles),  or 

a  -  0  =  B. 
Therefore 

a  =  lU  +  B), 
P  =  i{A-  B). 
Draw  RS  \\  to  EA.    ZEAR  =  ZARS  =  90°. 
By  similar  triangles 

BE/BR  =  AE/'SR 
^AE  .  8R 
AR  ■  AR 
But  BE  =  a  +  b  and  BR  =  a  -  b,  while 

AE      .  ,SR.„ 

-r-p-  =  tan  a  and  -j-k  =  tan  p. 
AR  AR 

Therefore  a+6  ^  tan  KA  + -B). 

inerelore  ^  _  ^      ^^^  ^^^  _  ^^ 

174.  The  following  special  formulas  are  readily  deduced  from  the 
sine  formulas  and  are  sometimes  useful  as  check  formulas  in  computa- 
tion. They  are  closely  related  to  the  law  of  tangents.  From  the 
proportion  ' 

a:b:c  =  sin  A: sin  B:sin  C 
by  composition  , 

c      _  sin  C 

0+6      sin  A  +  sin  B 

Now  by  §165  (1)  and  §167  (5)  this  may  be  written 

c 2  sin  jC  cos  jC 

0  +  6  ~  2  sin  UA  +  B)  cos  i(A  -  B)' 

Since  C  =  180°  -  (A  +  B),  therefore 

C/2  =  90°  -  |(A  +  B),  and  cos  C/2  =  sin  UA  +  B). 

c  sin  iC  cos  |(A  +  B)  ,,, 

a  +  b      cos  i(A  —  B)      cos  i(A  —  B; 

In  like  manner  it  can  be  proved  that 

c      _  sin  i(A  +  B)  ,2) 


a  -  b      sin  |(A  -  B) 
Both  (1)  and  (2)  can  be  readily  deduced  geometrically  from  Fig.  132. 
176.  The  s-formulas.    The  cosine  formula 

y 

a2  =  62  -}-  c^  -  26c  cos  A 


§175]  TRIGONOMETRIC  EQUATIONS  325 

can  be  written  in  the  forms 

a^  =  (b  +  cy  -  26c(l  +  cos  A),  (1) 

a2  =  (6  -  c)2  +  26c(l  -  cos  A),  (2) 

by  adding  (+26c)  and  (—26c)  to  the  right-hand  member  in  each 
case.    But  now  we  know  from  §166,  (1)  and  (2),  that 

1  +  C0S  A  =  2  cos"  (A /2), 
1  -  cos  JL  =  2  sm"  (A/2).  , 

Therefore  (1)  and  (2)  above  become 

o=  =  (6  +  c)2  -  46c  cos"  (A/2),  (3) 

'  a"  =  (6  -  cy  +  46c  sin"  (A/2).  (4) 

Writing  these  in  the  form      i\i 

ibc  sin"  (A/2)  =  o"]-  (6  -  c^,  (5) 

46c  cos"  (A/2)  =  (6  +  c)"  -  a",  (6) 

and  dividing  the  members  of  (5)  by  the  members  of  (6),  we 
obtain 

tan"  (A/2)  =1^1^.  (7) 

Factoring  the  numerator  and  denominator  we  obtain 

tan"  (A/2)  -fi  +  ^Tu^T^  +  l-  (8) 

'(6  +  c  +  o)  (6  +  c  —  a) 

Let  the  perimeter  of  the  triangle  be  represented  by  2s,  that  is, 
let 

a  +  6  +  c  =  2s. 

Hence,  subtracting  2c,  26,  and  2a  in  turn, 

a  +  6  —  c  =  2s  —  2c  (subtracting  2c), 
a  —  6-|-c  =  2s  —  26  (subtracting  26), 
6  +  c  —  a  =  2s  —  2a    (subtracting  2a). 

Therefore  equation  (8)  becomes 

tan"  (A/2)  =  (^ -/>)i^  '  <=) .  (g) 

s(s  —  a) 
Let 

(s  -  a)  (s  -  6)  (s  -  c)  /s  =  r\  (10) 


326        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§175 

Then 

or  ' 

Likewise 


tan^  (A/2)  =  r-V(s  -  aY, 
tan  (A/2)  =  r/(s  —  a). 


tan  (B/2)  =  r/(s  -  b), 
tan  (C/2)  =  r/(s  -  c), 


(11) 


(12) 
(13) 


Fig.  133. — Geometrical  derivation  of  the  s-formulas. 

Geometrically:  These  formulas  may  be  found  by  means  of  the 
diagram  Fig.  133.  Let  the  circle  0  be  inscribed  in  the  triangle  ABC; 
its  center  is  located  at  the  intersection  of  the  bisectors  of  the  internal 
angles  of  the  triangle.  Let  its  radius  be  r.  ATi  =  ATt,  BTt  =  BTz, 
CTi  =  CTi,  and  since  2s  =  a  +  6  +  c,  it  follows  that  one  way  of 
writing  the  value  of  s  is 

s  =  BTi  +  TiC  +  ATi. 


§176]  TRIGONOMETRIC  EQUATIONS  327 

Therefore 

ATi  =  s  -a.' 
Hence  it  follows  that 

tan  (A/2)  =  r/(s  -  o).  (14) 

Since  this  result  is  the  same  as  (11)  above,  it  proves  that  the  r  of 
equation  (10)  is  the  radius  of  the  inscribed  circle,  and  therefore  proves 
that  the  radius  of  the  inscribed  circle  may  be  expressed  by  the  formula 


Us  -  a)is  -6)(s  -c) 


a  fact  that  is  usually  proved  in  text  books  on  plane  geometry. 

176.*  Miscellaneous  Formulas  for  Oblique  Triangles.  The  fol- 
lowing formulas  are  given  without  proof.  They  are  occasionally 
useful  for  reference,  although  no  use  will  be  made  of  them  in 
this  book.  The  following  notation  is  used:  The  three  sides  of 
the  oblique  triangle  are  named  a,  b,  c,  and  the  angles  opposite 
these  A,  B,  C,  respectively.  The  semi-perimeter  of  the  triangle 
is  s,  OT  2s  =  a  +  b  +  c.  The  radius  of  the  circumscribed  circle 
is  B,  that  of  the  inscribed  circle  is  r,  and  the  radii  of  the  escribed 
circles  are  Ta,  n,  r^  tangent,  respectively,  to  the  sides  a,  b,  c 
of  the  given  triangle.    K  stands  for  the  area  of  the  triangle. 

s  =  4i?  cos  iA  cos  J-B  cos  §(7  (1) 

s  —  c  —  4Rsin  iA  sin  ^B  cos  iC  (2) 
and  analogs  for  s  —  a  and  s  —  b. 

r  =  iR  sin  JA  sin  iB  sin  iC  (3) 

Tc  =  4jB  cos  iA  cos  iB  sin  iC  (4) 
and  analogs  for  Ta  and  rt. 

Ta  =  s  tan  iA,n  =  s  tan  iB,  r^  =  s  tan  JC  (5) 

2K  =  ab  sinC  =  be  sin  A  =  ca  sin  B  (6) 

K  =  2R''  sin  A  sin  S  sin  C  =  |^  (7) 

K  =  Vsis  -a)  is-  b)  (s  -  c)  (8) 

K  =  rs  =  ra(s  —  a)  =  n(s  —  6)  =  r^is  —  c)  (9) 

Z2  =  rr^nr,  (10) 

K^  =  {s  -  a)  tan  iA  =  {s  -  b)  tan  iB  =  {s  -  c)  tan  |C  (11) 


328        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§177 

C.  NUMERICAL  SOLUTION  OF  OBLIQUE  TRIANGLES 

177.  An  oblique  triangle  possesses  six  elements;  namely,  the 
three  sides  and  the  three  angles.  If  any  three  of  these  six 
magnitudes  be  given  (except  the  three  angles),  the  triangle  is 
determinate,  or  may  be  constructed  by  the  methods  explained 
in  plane  geometry;  it  will  also  be  found  that  if  any  three  of  these 
six  magnitudes  be  given,  the  other  three  may  be  computed  by  the 
formulas  of  trigonometry,  provided,  that  the  given  parts  include 
at  least  one  side. 

It  is  convenient  to  divide  the  solution  of  triangles  into  four 
cases,  as  follows : 

I.  Given  two  angles  and  one  side. 
II.  Given  two  sides  and  an  angle  opposite  one  of  them. 

III.  Given  two  sides  and  the  included  angle. 

IV.  Given  the  three  sides. 

The  solution  of  these  cases  with  appropriate  checks  will  now 
be  given.  The  best  arrangement  of  the  work  of  computation 
usually  consists  in  writing  the  data  and  computed  results  in  the 
left  margin  of  a  sheet  of  ruled  letter  paper  (SJ  inches  X  11  inches) 
and  placing  the  computation  in  the  body  of  the  sheet.  Every 
entry  should  be  carefully  labeled  and  computed  results  should  be 
enclosed  in  square  brackets.  AU  work  should  be  done  on  ruled 
paper  and  invariably  in  ink.  Special  calculation  sheets  (forms 
M2  and  M7)  have  be'en  prepared  for  the  use  of  students.  Neat- 
ness and  systematic  arrangement  of  the  work  and  proper  checking 
are  more  important  thanr  rapidity  of  calculation. 

178.  Computer's  Rules.  The  following  computer's  rules  are 
useful  to  remember  in  logarithmic  work : 

Last  Digit  Even:  When  it  becomes  necessary  to  discard  a 
5  that  terminates  any  decimal,  increase  by  unity  the  last  digit 
retained  if  it  be  an  odd  digit,  but  leave  it  unchanged  if  it  be  an 
even  digit;  that  is,  keep  the  last  digit  retained  even.  Thus  log  tt 
=  0.4971;  hence  write  |  log  x  =  0.2486.  Also  log  sin  18°  5' 
=  9.4900  +  (correction)  19.5  =  9.4920. 

Of  course  if  the  discarded  figure  is  greater  than  5,  the  last 
digit  retained  is  increased  by  1,  whUe  if  the  discarded  figure  is 
less  than  5,  the  last  digit  retained  is  unchanged. 


8179] 


TRIGONOMETRIC  EQUATIONS 


329 


Functions  or  Angles  in  Second  Quadbant:  In  finding 
from  the  table  any  function  of  an  angle  greater  than  100°  (but 
<  180°)  replace  the  first  two  digits  of  the  number  of  degrees  in  the 
angle  by  their  sum  and  take  the  co-function  of  the  result.  The 
method  is  valid  because  it  is  equivalent  to  the  subtraction 
of  90°  from  the  angle.  By  §163  this  always  gives  the  cor- 
rect numerical  value  of  the  function.  The  algebraic  sign  should 
be  taken  into  account  separately.  Thus,  sin  157°  32'  7"  = 
cos  67°  32'  7".  In  case  of  an  angle  between  90°  and  100°, 
ignore  the  first  digit  and  proceed  in  the  same  way.    Thus, 

tan  97°  57'  42"  =  -  cot  7°  57'  42" 

179.  Case  I.    Given  two  angles  and  one  side,  as  A,  B,  and  c. 

1.  To  find  €,  use  the  relation  A+B  +  C  =  180°. 

2.  To  find  a  and  6,  use  the  law  of  sines,  §171. 

3.  To  check  results,  apply  the  check  formula  (1)  or  (2)  §174. 

Illtjstkation:  In  an  oblique  triangle,  let  c  =  1492,  A  =  49°  52', 
B  =  27°  15'.     It  is  required  to  compute  C,  a,  b. 

The  following  form  of  work  is  self  explanatory.  It  should  be  noted 
that  the  process  of  work  and  the  meaning  of  each  number  entering  the 
calculation  is  properly  indicated  or  labeled  in  the  work. 


Numerical  Work 

Given 

To  find  o,  b,  and  C. 

c    =  1492 

Formulas 

A  =  49°  52' 

C  =  180  -  U  +  B)  = 

=  (102°  53') 

B  =  27°  15' 

c  sin  A 

Work. 

"  ~      sin  C 
,  _  c  sin  B 
sin  C 

log  sin  A  = 
log  c          = 

9.8834  - 
3.1738 

■  10 

log  sin  B  = 

9.6608  - 

-  10 

^ 

log  sin  C  = 

9.9889  - 

-  10 

log  o 

3.0683 

log  6 

2.8457 

a         = 

[1170.] 

6 

[701.] 

Check. 

Check  Formula 

c  -b     = 

:  791 

a             sin  i(C  +  B) 

C  +  B  = 

:  130°    8' 

c 

-  h          sin  i(C  -  B) 

C  -  B  ^ 

:    75°  38' 

330         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§I80 


i(C  +  B)  =  65°    4' 

UC  -  B)  =  37°  49' 

log  a  =  3.0683 

log  c  -  6  =  2.8982 

log  r^b         =  01701 

log  sin  i  {C  +  B)  =  9.9575  -  10 
log  siD  i  (e  -  B)  =  9.7875  -  10 

Examples 

Find  the  remaining  parts,  given : 

1.  A  =  47°  20',  B  =  32°  10', 

2.  B  =  37°  38',  C  =  77°  23', 

3.  B  =  25°  2',  C  =  105°  17', 

4.  C  =  19°  35',  A  =  79°  47', 


Check 


a  =  739. 
6  =  1224. 
6  =  0.3272. 
c  =  56.47. 


180.  Case  n.    Given  two  sides  and  an  angle   opposite  one  of 
hem,  as  a,  b,  and  A . 


Fig.    134. — Case    II   of    triangles,    for    one,   two,   and    impossible 
solutions. 

1.  To  find  B,  use  the  law  of  sines,  §171. 

2.  To  find  C,  use  the  equation  A  +B  +  C  =  180°. 

3.  To  find  c  use  the  law  of  sines. 

4.  To  check,  apply  the  check  formula  (I)  or  (2),  §174. 

When  an  angle  as  B,  above,  is  determined  from  its  sine,  it  admits 
of  two  values,  which  are  supplementary  to  each  other.  There 
may  be,  therefore,  two  solutions  to  a  triangle  in  Case  II.  The 
solutions  are  illustrated  in  Fig.  134. 


§180] 


TRIGONOMETRIC  EQUATIONS 


331 


•  In  case  one  of  the  two  values  of  B  when  added  to  the  given 
angle  A  gives  a  sum  greater  than  two  right  angles,  this  value 
of  B  must  be  discarded,  and  but  one  solution  exists.  If  a  be 
less  than  the  perpendicular  distance  from  C  to  c,  no  solution 
is  possible. 

Illustration:    Solve   the    triangle  if   a  =  345,    6  =  534,   and 
A  =  25°  25'. 
The  solution  is  readily  understood  from  the  following  work. 

Numerical  Work 
Given 

a  =  345 
6  =  534 


To  find  c,  B,  and  C. 
Formulas 

.     „       b  sin  A 


A  =  25°  25' 

a 

Work. 

C  =  180  -  (A  +  B) 

log  6          =2.7275 

a  Bin  C 

log  sin  jl  =  9.6326  -  10 

sin  A 

log  a          =2.5378 

logsmB  =  9.8223  -  10 

B  =  [41°37'.l] 

or 

[138°  22'. 9] 

A+B      =67°2'.l 

163°  47'. 9 

C  =  [112°  57'. 9] 

or 

[16°12'.l] 

logo          =2.5378 

2.5378 

log  sin  C  =  9.9641  -  10 

9.4456  -  10 

log  sin  A  =  9.6326  -  10 

9.6326  -  10 

log  c          =  2.8693 

2.3508 

c  =  [740.1] 

[224.3] 

Check 

6          sin  i(C  +  A)       ^ 

a       _  sin  K-B  +  C) 

c  —  a      sm  t 

c  —  a 

C  +  A 

C  —  A 

W  +  A) 

KC  -  A) 

log  6 

log  (c  -  o) 

logQ 

log  sin  KC  + 

log  sin  KC  — 

logO 


iC  -A) 
=  395.1 
=  138°  22'. 
=    87°  32'. 
=    69°  11' 
=    43°  46' 
=  2.7275 
=  2.5967 
=  0.1308 
=  9.9707  - 

A)  =  9.8400  - 
=  0.1307 


A) 


10 
10 


b  —  c      sin  |(B 

b  -c 

B  +  C 

B  -C 

UB  +  C) 

i(B  -  C) 

log  o 

log  (6  -  c) 

logQ' 

Iogsini(B  +  C) 

logsini(fi-C) 

logQ' 


=  309.7 
=  154°  35' 
=  122°  10'.  8 
=    77°  17'.  5 
=    61°    5'. 4 
=  2.5378 
=  2.4910 
=  0.0468 
=  9.9892  - 
=  9.9422  - 
=  0.0470 


10 
10 


332         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§181 

Examples 
Compute  the  unknown  parts  in  each  of  the  following  triangles : 

,  1.  a  =  0.8,  b  =  0.6,  B  =  40°  15'. 

2.  o  =  8.81,  6  =  11.87,  A  =  19°  9'. 

3.  6  =  81.05,  c  =  98.75,  C  =  99°  19'. 

4.  c  =  50.37,                   a.  =  58.11,  C  =  78°  13'. 
6.  a  =  1213,                     6  =  1156,  B  =  94°  15'. 

181.  Case  III.    Given  two  sides  and  the  included  angle,  as  a,b,C. 

1.  To  find  A  +B,useA  +B  =  180°  -  C. 

2.  To  find  A  and  B,  compute  (A  —  B)/2  by  the  law  of  tangents, 
§173,  equation  (4),  then  A  =  (A  +  B)/2  +  (A  -  B)/2  and 
B  =  (A+  B)/2  -  {A  -  B)/2. 

3.  To  find  c,  use  law  of  sines,  §171. 

4.  To  check,  use  law  of  sines. 

Illustration:    Given  a  =  1033,  6  =  635,  C  =  38°  36'. 


Numerical  Work 

Given 

To  find  c.  A,  and  B. 

a  = 

1033                                      Formulas 

b  = 

635                                    A+B  =  180  -C  =  141°  24' 

C  = 

3*°  3^'                      tan  UA-B)  =  ^-=-^  tanJCA  +  B) 

o'sin  C 

tsin  A 

Work 

a  -b                  '   =    398 

a  +  b                      =  1668 

\{A+B)               =  70°  42' 

log  (a  -  6)             =  2.5999 

logtanKA+'B)   =0.4557 

-     log(o  +  6)             =3.2222 

log  tan  4U  -B)   =  9.8334  -  10 

1{A  -B)               =  34°  16.3' 

A                            =[104°  58.3'] 

B                            =    [36°  25.7'] 

logo                       =3.0141 

log  sin  C                =  9.7951  -  10 

log  sin  A                =  9.9850  -  10 

logo                        =2.8242 

c  =  [667.1] 

§182] 


TRIGONOMETRIC  EQUATIONS 


333 


Check 


b  sin  C 


sin  B 
log  6  =2.8028 

log  sin  C  =  9.7951  -  10 
log  sin  B  =  9.7737  -  10 
logc  =2.8242 

c  =  [667.1] 

Examples 

Compute  the  unknown  parts  in  each  of  the  following  triangles : 

1.  a  =78.9,  6=68.7,  C  =  78°  10'. 

2.  c  =  70.16,  a  =  39.14,  B  =  16°  16'. 

3.  6  =  1781,  c  =  982.7,  A  =  123°  16'. 

182.  Case  IV.    Given  the  three  sides. 

1.  To  find  the  angles,  use  the  s-formulas,   §175,   (11), 
and  (13). 

2.  To  check,  use  A  +  B  +  C  =  180°. 

Illustration:  Given  a  =  455,  6  =  566,  c  =  677,  find  A,  B  and  C. 
Numerical  Work 


(12) 


Given 


"Work. 


a  = 

455 

6  = 

566 

c  = 

677 

2s  = 

1698 

s  = 

849 

s 

—  a  = 

394 

s 

-6  = 

283 

s 

—  c  = 

172 

To  find  A,  B  and  C. 
Formulas 

T 

tan  iA  = ' 


tan  hB  = 


tan  JC  = 


s  -  b 
r 


where  r  =  ^'(^  -  aKs  -  bHs  -  e). 

2si  =  1698 
log  (s-a)  =  2.5955 
log  (s-b)  =  2.4518 
log  (s  -  c)  =  2.2355 
logs  =2.9289 

logr"  =4.3539 

logr  =2.1770 

log  tan  JA  =  9.5815  -  10 
log  tan  IB  =  9.7252  -  10 
log  tan  iC  =  9.9415  -  10 

•  Adding   th«   four  numbera  above  this  line  cheoks  the  subtractions    (>  — a), 
{»  -h),  etc. 


334         ELEMENTARY  MATHEMATICAL  ANALYSIS      (§182 

iA  =  20°  53' 
JJS  =  27°  58' 
JC  =  41°  9' 
A  =  [41°  46'] 
B  =  [55°  56'] 
C  =  [82°  18'] 
Check. 

A+B+C    =  180° 

Exercises 

Find  the  values  of  tlJe  angles  in  each  of  the  following  triangles : 

1.  a  =  173,  6  =  98.6,  c  =  230. 

2.  a  =  8.067,  6  =  1.765,  c  =  6.490. 

3.  a  =  1911,  6  =  1776,  c  =  1492. 

Miscellaneous  Problems 

The  instructor  will  select  only  a  limited  number  of  the  following 
problems  for  actual  computation  by  the  student.  The  student 
should  be  required,  however,  to  outline  in  writing  the  solution  of  a 
number  of  problems  which  he  is  not  required  actually  to  compute,  and, 
when  practicable,  to  block  out  a  suitable  check  for  each  one  of  them. 

1.  From  one  corner  P  of  a  triangular  field  PQB  the  side  PQ  bears 
N.  10°  E.  100  rods.  QR  bears  N.  63°  E.  and  PR  bears  N.  38°  10'  E. 
Find  the  perimeter  and  area  of  the  field. 

2.  The  town  B  lies  15  miles  east  of  A,  C  lies  10  miles  south  of  A. 
X  lies  on  the  Hne  BC,  and  the  bearmg  of  AX  is  S.  46°  20'  E.  Find 
the  distances  from  X  to  the  other  three  towns. 

3.  To  find  the  length  of  a  lake  (Fig.  135),  the  angle  C  =  48°  10',  the 
side  a  =  4382  feet,  and  the  angle  B  =  62°  20'  were  measured.  Find 
the  length  of  the  lake  c,  and  check. 

4.  To  continue  a  line  past  an  obstacle  L,  Pig.  136,  the  line  BC  and 
the  angles  marked  at  B  and  C  were  measured  and  found  to  be  1842 
feet,  28°  15',  and  67°  24',  respectively.  Find  the  distance  CD,  and 
the  angle  at  D  necessary  to  continue  the  line  AB;  also  compute  the 
distance   BD. 

5.  Find  the  longer  diagonal  of  a  parallelogram,  two  sides  being 
69.1  and  97.4  and  the  acute  angle  being  29°  34'. 

What  is  the  magnitude  of  the  single  force  equivalent  to  two  forces 
of  69.1  and  97.4  dynes  respectively,  making  an  angle  of  29°  34'  with 
each  other? 

6.  A  force  of  75.2  dynes  acts  at  an  angle  of  35°  with  a  force  F. 
Their  resultant  is  125  dynes.    What  is  the  magnitude  of  Fl 


§182] 


TRIGONOMETRIC "  EQUATIONS 


335 


7.  The  equation  of  a  circle  is  p  =  10  cos  6.  The  points  A  and 
B  on  this  circle  have  vectorial  angles  31°  and  54°  respectively.  Find 
the  distance  AB,  (1)  along  the  chord;  (2)  along  the  arc  of  the  circle. 

8.  Knd  the  lengths  of  the  sides  of  the  triangle  enclosed  by  the 
straight  lines : 


e  =  26° 


115°;  p  cos  (9  -  45°)  =  50. 


Fig.  135. — Diagram  for 
Problem  3. 


Fig.  136. — Diagram  for  Problem  4. 


9.  A  gravel  heap  has  a  rectangular  base  100  feet  long  and  30  feet 
wide.  The  sides  have  a  slope  of  2  in  5.  Find  the  number  of  cubic 
yards  of  gravel  in  the  heap. 

10.  A  point  B  is  invisible  and  inaccessible  from  A  and  it  is  necessary 
to  find  its  distance  from  A.  To  do  this  a  straight  line  is  run  from  A 
to  P  and  continued  to  Q  such  that  B  is  visible  from  P  and  Q.  The 
following  measurements  are  then  taken:  AP,  =  2367  feet;  PQ  =  2159 
feet;  APB  =  142°  37'.3;  AQB  =  76°  13'.8.     Find  AB. 

11.  To  determine  the  height  of  a  mountain  the  angle  of  elevation 
of  the  top  was  taken  at  two  stations  on  a  level  road  and  in  a  direct 
line  with  it,  the  one  5280  yards  nearer  the  mountain  than  the  other. 
The  angles  of  elevation  were  found  to  be  2°  45'  at  the  further  station 
and  3°  20'  at  the  nearer  station.  Find  the  horizontal  distance  of  the 
mountain  top  from  the  nearer  station  and  the  height  of  the  mountain 
above  it.     Use  S  and  T  functions. 

12.  Explain  how  to  find  the  distance  between  two  mountain  peaks 
Ml  and  Af  2,  (1)  when  A  and  B  at  which  measurements  are  taken  are  in 
the  same  vertical  plane  with  Mi  and  M^;  (2)  when  neither  A  nor  B 
is  in  the  same  vertical  plane  with  Mi  and  M2. 

13.  The  sides  of  a  triangular  field  are  534  yards,  679  yards,  and  474 
yards.  The  first  bears  north,  and  following  the  sides  in  the  order  here 
given  the  field  is  always  to  the  left.  Find  the  bearing  of  the  other 
two  sides 'and  the  area. 


336        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§182 

14.  From  a  triangular  field  whose  sides  are  124  rods,  96  rods,  and 
104  rods  a  strip  containing  10  acres  is  sold.  The  strip  is  of  uniform 
width,  having  as  one  of  its  parallel  sides  the  longest  side  of  the  field. 
Knd  the  width  of  the  strip. 

16.  Three  circles  are  externally  mutually  tangent.  Their  radii  are 
5,  6,  and  7  feet.    Find  the  area  and  perimeter  of  the  three-cornered 

area  enclosed  by  the  circles  and  the 
length  of  a  wire  that  will  enclose  the 
group  of  three  circles  when  stretched 
about  them. 

16.  To  find  the  distance  between  two 
inaccessible  objects  C  and  D,  Kg.  137, 
two  points  A  and  B  are  selected  from 
which  both  objects  are  visible.    The  dis- 

137_ Diagram    for    tS'^ce   AB   is   found   to   be   7572   feet. 

Problem  16.  The  following  angles  were  then  taken: 

ABD  =  122°  37'  BAC  =    80°  20' 

ABC  =    70°  12'  BAD  =    27°  13' 

Knd  the  distance  DC  and  check. 

17.  A  circle  of  radius  o  has  its  center  at  the  point  (pi,  9i).  Knd  its 
equation  in  polar  coordinates.     (Use  law  of  cosines.) 

18.  A  surveyor  desired  the  distance  of  an  inaccessible  object  0 
from  A  and  B,  but  had  no  instruments  to  measure  angles.  He 
measured  AA'  in  the  Une  AO,  BB'  in  the  line  BO;  also  AB,  BA',  and 
AB'.     How  did  he  find  OA  and  OB? 

19.  From  a  point  A  a  distant  object  C  bears  N.  32°  16'  W.  and 
from  B  the  same  object  bears  N.  50°  W.  AB  bears  N.  10°  39'  W. 
The  distance  AB  is  1000  yards.     Knd  the  distance  AC. 

20.  The  angle  of  elevation  of  a  mountain  peak  is  observed  to  be 
19°  30'.  The  angle  of  depres.sion  of  its  image  reflected  in  a  lake  1250 
feet  below  the  observer  is  found  to  be  34°  5'.  Find  the  height  of  the 
mountain  above  the  observer  and  the  horizontal  distance  to  it.  (See 
Fig.  138.) 

21.  One  side  of  a  mountain  is  a  smooth  eastern  slope  inclined  at  an 
angle  of  26°  10'  to  the  horizontal.  At  a  station  A  a  vertical  shaft  is 
sunk  to  a  depth  of  300  feet.  From  the  foot  of  the  shaft  two  horizontal 
tunnels  are  dug,  one  bearing  N.  22°  30'  E.  and  the  other  S.  65°  E. 
These  tunnels  emerge  at  B  and  at  C  respectively.  Find  the  lengths 
of  the  tunnels  and  the  lengths  of  the  sides  of  the  triangle  ABC. 

22.  A  rectangular  field  ABCD  has  side  AB  =  40  rods;  AD  =  80 
rods.  Locate  a  point  P  in  the  diagonal  AC  so  that  the  perimeter  of 
the  triangle  APB  will  be  160  rods.  {Hint:  Express  perimeter  as  a 
function  of  angle  at  P.) 


§182] 


TRIGONOMETRIC  EQUATIONS 


337 


8.  Find  the  area  enclosed  by  the  lines  y  =  k'  y  =  \/3  x,  and  the 


Fig.  138. — Diagram  for  Problem  20. 

circle  x'  —   lOs  +  ^^^  =  0.     (Hint:    Change  to  polar  coordinates.) 

24.  The  displacement  of  a  particle  from  a  fixed  point  is  given  by 

d  =  2.5  cos  t  +  2.5  sin  t. 

What  values  of  t  give  maximum  and  minimum  displacements;  what 
is  the  maximum  displacement? 

25.  A  quarter  section  of  land  is  enclosed  by  a  fence.  A  farmer 
wishes  to  make  use  of  this  fence  and  60  rods  of  additional  fencing  in 
making  a  triangular  field  in  one  comer  of  the  original  tract.  Find 
the  field  of  greatest  possible  area.  Show  that  it  is  also  the  field  of 
maximum  perimeter,  under  the  conditions  given. 

26.  A  force  Fi  =  100  dynes  makes  an  angle  of  6°  with  the  horizontal, 
and  a  second  force  Fi  =  50  dynes  makes  an  angle  of  90°  with  Fi. 
Determine  B  so  that  (1)  the  sum  of  the  horizontal  components  of 
Fi  and  Ft  shall  be  a  maximum;  (2)  so  that  the  sum  of  the  vertical  com- 
ponents shall  be  zero. 

27.  Find  the  area  of  the  largest  triangular  field  that  can  be  enclosed 
by  200  rods  of  fence,  if  one  side  is  70  rods  in  length. 

22 


338         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§182 

28.  Change  the  equation  of  the  curve  xy  =  I  to  polar  coordinates, 
rotate  through  —  45°,  and  change  back  to  rectangular  coordinates. 

29.  A  particle  moves  along  a  straight  line  so  that  the  distance 
varies  directly  as  (sin  t  +  cos  t).  When  t  =  7r/4,  the  distance  is  10. 
Find  the  equation  of  motion. 

30.  From  the  top  of  a  lighthouse  60  feet.high  the  angle  of  depression 
of  a  ship  at  anchor  was  observed  to  be  4°  52';  from  the  bottom  bf  the 
lighthouse  the  angle  was  4°  2'.  Required  the  horizontal  distance  from 
the  lighthouse  to  the  ship  and  the  height  of  the  base  of  the  lighthouse 
above  the  sea. 

31.  The  Une  AB  runs  north  and  south.  The  line  AC  makes  an 
angle  of  52°  8'. 6  with  AB.  Locate  the  Une  BC  perpendicular  to  AB 
so  that  the  area  ABC  shall  be  1  acre. 

32.  University  Hall  casts  a  shadow  324  feet  long  on  the  hillside 
on  which  it  stands.  The  slope  of  the  hillside  is  15  feet  in  100  feet, 
and  the  elevation  of  the  sun  is  23°  27'      Find  the  height  of  the  building. 

33.  To  determine  the  distance  of  a  fort  A  from  a  place  B,  a  line  BC 
and  the  angles  ABC  and  BCA  were  measured  and  found  to  be  3225.5 
yards,  60°  34',  and  56°  10'  respectively.     Find  the  distance  AB. 

34.  A  balloon  is  directly  over  a  straight  level  road,  and  between  two 
points  on  the  road  from  which  it  is  observed.  The  points  are  15,847 
feet  apart,  and  the  angles  of  elevation  are  49°  12'  and  53°  29'.  Find 
the  height. 

35.  Two  trees  are  on  opposite  sides  of  a  pond.  Denoting  the  trees 
by  A  and  B,  we  measure  AC  =  297.6  feet,  BC  =  864.4  feet,  and  the 
angle  ACB  =  87°  43'.     Find  AB. 

36.  Two  mountains  are  9  and  13  miles  respectively  from  a  town, 
and  they  include  at  the  town  an  angle  of  71°  36'.  Find  the  distance 
between  the  mountains. 

37.  The  sides  of  a  triangular  field  are,  in  clockwise  order,  534  feet, 
679  feet,  and  474  feet;  the  first  bears  north;  find  the  bearings  of  the 
other  sides  and  the  area. 

38.  Under  what  visual  angle  is  an  object  7  feet  long  seen  when  the 
eye  is  15  feet  from  one  end  and  18  feet  from  the  other? 

39.  The  shadow  of  a  cloud  at  noon  is  cast  on  a  spot  1600  feet  west 
of  an  observer,  and  the  cloud  bears  S.,  76°  W.,  elevation  23°.  Find 
its  height. 


CHAPTER  XI 


SIMPLE  HARMONIC  MOTION  AND  WAVES 

183.  Simple  Harmonic  Motion.  In  Fig.  139,  x  =  0T>  = 
a  cos  DOM,  where  a  is  the  radius  of  the  circle.  If  now  the  point 
M  is  thought  of  as  moving  with  constant  or  uniform  speed  on  the 
circle,  starting  at  A,  or  (which  amounts  to  the  same  thing)  if 
the  radius  OM  is  thought  of  as  moving  with  constant  angular 
velocity,  say  k  radians  per  second,  starting  from  OA,  then  angle 
DOM  =  kt  and  the  position  of  the  point  D  at  time  t  is  given  by 

X  =  a  cos  kt,  (1) 

where  t  is  the  time  in  seconds  required  for  OM  to  move  from  posi- 
tion OA  to  position  OM. 

Let  us  study  the  motion  of  the  point  D  as  M  moves  on  the 
circle  with  constant  speed. 
D  starts  at  A  and  moves  to 
the  left  with  increasing  speed 
until  it  arrives  at  ■  0,  where 
its  speed  begins  to  decrease, 
decreasing  to  0  at  A'.  Then 
the  point  moves  to  the  right 
with  increasing  speed  until  it 
again  passes  through  0,  after 
which  its  speed  diminishes, 
becoming  0  when  it  arrives 
at  A .  Then  the  whole  motion 
is  repeated.  A  body  whose 
position  on  a  straight  line  is 
given  at  any  instant  by  an  equation  of  the  form  (1),  that  is  one 
which  moves  as  the  point  D  does,  is  said  to  describe  simple  harmonic 
motion.  On  account  of  the  frequency  with  which  this  term  will 
occur,  we  shall  abbreviate  it  by  the  symbols  S.H.M.    Examples 

.339 


Fig.  139. 


340        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§183 

of  bodies  that  move  approximately  in  this  way  are:  The  bob  of  a 
pendulum,  a  point  in  the  prong  of  a  vibrating  tuning  fork,  a 
point  in  a  vibrating  violin  string,  the  particles  of  air  during  the 
passage  of  a  sound  wave.  The  motion  is  oscillatory  in  character 
and  repeats  itself  in  definite  intervals  of  time. 

The  length  of  this  interval  can  be  easily  found  by  considering 
the  motion  of  the  point  M  on  the  circle.  The  point  D  starting 
from  any  given  position  will  return  to  this  position  moving  in  the 
same  direction  after  an  interval  of  time  which  is  the  time  required 
for  M  to  describe  the  circle,  i.e.,  after  2ir/k  seconds,  the  time  in 
which  the  radius  OM  describes  the  angle  2ir  radians  at  the  rate  of 
k  radians  per  second.  This  time  within  which  a  body  executing 
S.H.M.  performs  a  complete  oscillation  is  called  the  period  of  the 
S.H.M.    It  is  denoted  by  T.    Thus 

T  =  ^.  (2) 

This  expression  can  be  obtained  directly  from  the  equation  x  = 
a  cos  kt  by  means  of  the  fact  that  the  cosine  is  a  periodic  function 
of  period  2x.  The  period  T  is  the  amount  by  which  t  must  be 
increased  in  order  to  increase  the  angle  kt  by  the  amount  27r. 
If  t  be  increased  by  the  amount  2ir/k,  then  kt  is  increased  by 
2x,  because 

fc(t+^)  =A;i  +  2ir. 
The  number  of  complete  periods  per  second  is 

^  =  T  =  .V  (3) 

N  is  called  the  frequency  of  the  S.H.M. 

'if  instead  of  counting  time  from  the  instant  at  which  the 
auxiliary  point  M  passed  through  A,  we  count  it  from  the  instant 
it  passed  through  E,  then  ZEOM  =  kt,  and  it  is  clear  that 
ZAOM  =  (kt  —  e)  if  e  stands  for  the  constant  angle  EOA. 
Then  (1)  becomes 

X  =  a  cos  (Jet  —  «).  (4) 

The  number  a  is  called  the  amplitude,  e  is  called  the  epoch  angle, 


§184]      SIMPLE  HARMONIC  MOTION  AND  WAVES         341 

and  (Jet  —  e)  is  called  the  phase  angle  of  the  S.H.M.  represented 
by  (4). 

In  like  manner  the  point  D2,  the  projection  of  the  point  M  upon 
the  vertical  diameter  of  the  circle  in  Fig.  139,  describes  S.H.M. 
Its  equation  is 

2/  =  a  sin  (kt  —  e),  (5) 

where  time  t  is  measured  from  the  instant  M  passes  through  E. 

184.  Mechanical  Generation  of  S.H.M.  Fig.  140  illustrates 
a  way  in  which  S.H.M.  may  be  described  by  mechanical  means. 


rp 

^ 

- 

B 

—3"             \r\\ 

^ 

^. 

< 

^4^^  ho. 

C                         K 

1 

^ 

\ 



2?__"5 =__ 

>  s' 

—L 

7L S 

'^ :  -  - 

-  ,7    _      _    _  _      s_ . 

'-'-1 

s, 

7                                    ■S 

s  :,                       , 

^      €.                                                            S           ., 

_  Si^:                    cj- 

7                                           _S    ^1 

\ 

ii 

_^ 

c  1   S                                    ' 

'                                                  ^ 

~-~. 

■s                                   ? 

v^ 

"5                                     7' 

\?              rs                   M 

"--III  II  llml  II 1  WyW  II 

H 

■^ 

|LJ_LiJ_LJ_LJ_ri    I  ^                1 

Pig. 


140. — Mechanical  generation   of    simple    harmonic    motion, 
and  of  a  simple  progressive  wave. 


Let  the  uniformly  rotating  wheel  OAB  be  provided  with  a  pin 
M  attached  to  its  circumference  and  free  to  move  in  the  slot 
of  the  cross-head  as  shown,  the  arm  "of  the  cross-head  being  re- 
stricted to  vertical  motion  by  suitable  guides  G-G\-  Then,  as  the 
wheel  rotates,  any  point  P  of  the  arm  of  the  cross-head  describes 
simple  harmonic  motion  in  a  vertical  direction.  The  amplitude 
of  the  S.H.M.  is  the  radius  of  the  circle,  or  OB;  its  period  is  the 
time  required  for  one  complete  revolution  of  the  wheel. 


342         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§18S 

Exercises 

1.  Find  the  periods  of  the  following  S.H.M. : 

{a)  y  =  3  sin  2t.  (e)  y  =  a  sin  (10<  —  7r/3). 

(6)  2/  =  10  sin  (1/2) «  (/)  t/  =  o  sin  (2«/3  -  27r/5). 

(c)  y  =  7  cos  4<.  ig)  y  =  a  sin  (6<  +  c). 
{d)  y  =  a  sin  27r<. 

2.  Give  the  amplitudes  and  epoch  angles  in  each  of  the  instances 
given  in  exercise  1. 

3.  The  bob  of  a  second's  pendulum  swings  a  maximimi  of  4  cm. 
each  side  of  its  lowest  position.  Considering  the  motion  as  rectilinear 
S.H.M.,  write  the  equation  of  motion.' 

^  The  term  period  is  used  differently  in  the  case  of  a  pendulum  than  in  the  case 
of  S.H.M.  The  time  of  a  swing  is  the  period  of  a  pendulum;  the  time  of  a  awino- 
swang  is  the  period  of  a  S.H.M. 

Write  the  equation  of  motion  of  a  pendulum  of  the  same  length 
which  was  released  from  the  end  of  its  swing  1/2  second  after  the  first 
pendulum  was  similarly  released. 

4.  A  particle  moves  in  a  straight  Une  in  such  a  way  that  its  dis- 
placement from  a  fixed  point  of  the  line  is  given  by  d  =  2  cos*  t.  Show 
that  the  particle  moves  in  S.H.M.,  and  find  the  amplitude  and  period 
of  the  motion. 

6.  A  particle  moves  in  a  vertical  circle  of  radius  2  units  with  angular 
velocity  of  20  radians  per  second.  Counting  time  from  the  instant 
the  particle  was  at  its  lowest  position,  write  the  equation  of  motion 
of  its  projection  (1)  upon  the  vertical  diameter;  (2)  upon  the  horizon- 
tal diameter;  (3)  upon  the  diameter  bisecting  the  angle  between  the 
horizontal  and  vertical. 

186.  S.H.M.  Record  on  Smoked  Glass.  If  P,  Fig.  140,  be  a 
tracing  point  attached  to  the  vertical  arm  of  the  cross-head  and 
capable  of  describing  a  curve  on  a  piece  of  smoked  glass,  HK,  which 
is  moved  to  the  right  at  constant  speed,  then  when  P  describes 
S.H.M.  in  the  vertical  line  OP,  the  curve  NiCTNJ'  traced  on  the 
plate  HK  is  a  sinusoid.  For,  if  iVj  be  taken  as  origin,  and  if  for 
convenience  positive  abscissas  be  measured  to  the  left,  the  coordi- 
nates of  P  are 

X  =  Vt, 

and  y  =  a  sin  {kt  —  e) 


§186]      SIMPLE  HARMONIC  MOTION  AND  WAVES         343 

where   V  is  the  linear  velocity  of  the  plate.    Eliminating  t 
between  these  two  equations, 

y  =  asin  yy  -  ej  (1) 

the  equation  of  a  sinusoid. 

If  the  plate  HK  moves  with  the  same  velocity  as  the  point  M, 
we  have 

V  =  ha 
and  equation  (1)  becomes 

-  =  sm  -,  (2) 

a  a 


the  equation  of  an  undistorted  sinusoid.' 

186.*    Composition  of  Two  S.H.M.'s  at  Right  Angles. 

It  is  obvious  that 

X  =  a  cos  ht 

represents  a  S.H.M.  one  quarter  of  a  period  in  advance  of  a;'  =  a  sin  kt, 

since   sin  I  fci  +  ^1    =  cos  kt.     A  pair  of  S.H.M.'s  possessing  this 

property  are  said  to  be  in  quadrature.     (4)  and  (5),  §183,  may  be  said 
to  be  in  quadrature. 

We  have  shown  that  if  a  point  M,  moving  uniformly  on  a  circle,  be 
projected  upon  both  the  X-  and  7-axes,  two  S.H.M.'s  result.  The 
phase  angles  of  these  two  motions  differ  from  each  other  by  7r/2. 
The  converse  of  this  fact,  namely  that  uniform  motion  in  a  circle  may 
be  the  resultant  of  two  S.H.M.'s  in  quadrature,  is  easily  proved,  for  the 
two  equations  of  S.H.M. 

X  =  a  cos  kt 

y  =  a  sin  kt 

are  obviously  the  parametric  equations  of  a  circle.  Hence  the  theorem  : 
Uniform  motion  in  a  circle  may  he  regarded  as  the  residtant  of  two 

S.H.M.'s  of  equal  amplitudes  and  equal  periods  and  differing  by  7r/2  in 

phase  angle. 
This  important  truth  is  illustrated  by  Fig.  141.    Let  the  X-  and 

1  The  student  should  note  that  ^  =  sin  -  is  of  exactly  the  same  shape  as  y  =  sm  x, 

for  multiplying  both  ordinates  and  abscissas  of  any  curve  by  a  is  merely  constructing 

the  curve  to  a  different  scale.     However,  ^  =  sin  o  is  a  distorted  sinusoid,  for  the 

ordinates  of  y  =  sin  x  are  multiplied  by  3  while  the  abscissas  are  multiplied  only 
by  2. 


344        ELEMENTARY  MATHEMATICAL  ANALYSIS     [§187 


y-axes  be  divided  proportionally  to  the  trigonometric  sine,  as  in  Fig. 
59.  Through  the  points  of  division  of  the  two  axes  draw  lines  per- 
pendicular to  the  axes,  thus  dividing  the  plane  into  a  large  number  of 
small  rectangles.  Starting  at  the  end  of  one  of  the  axes,  and  sketch- 
ing the  diagonals  of  successive  cornering  rectangles,  the  circle  ABA'B' 
is  drawn. 

If  the  same  construction  be  carried  out  for  the  case  in  which  the  Y- 
axis  is  divided  proportionally  to  6  sin  kt  and  in  which  the  X-axis  is 

divided  proportionally  to  osin 
kt,  the  ellipse  AiBiA'iB'i  re- 
sults. These  facts  are  merely 
a  repetition  of  the  statements 
made  in  §84. 

187.  Waves.— Let  Fig.  142 
represent  a  section  obtained 
by  passing  at  any  instant  a 
vertical  plane  perpendicular 
to  the  crests  of  a  series  of 
small  waves  on  the  surface 
of  a  body  of  water.  The 
wavy  line  represents  the  ap- 
pearance of  the  surface  at 
any  instant.  It  is  a  fact 
that  its  equation  is,  in  the 
case  of  small  waves  or  ripples, 


■ 

1 , 

B 

-- 

-7- 

— 

' — 

— 

— 

^s-:: 

7 

^   S 

/ 

S 

/ 

^ 

, 

0 

\ 

\ 

1 

\ 

\ 

7 

A 

7   _ 

.__ 

— 

— 

— 

— 

^2-: 

■p 

ff\r 

— 

— 

— 

^ 

V--S- 

— F 

— 

— 

— 

— 

g — 5, 

n' 

1 

Ai    -- 

[ 

1 

, 

■-: 

.  =  -^ 

1 

1 

i--^-^- 

Ax 


2/  =  a  sin  he. 


(1) 


Fig.  142  represents  the  seo- 
■^1  tion  of  the  surface  at  any 

Fig.    141. — The    circle   and   the     instant,    say    t  =  0.      Now 

eUipse  considered  as  generated  by     „„„i     „,„„„„  „„„„    c a 

two  S.H.M.'s  in  quadrature.  ^'^.^f    ^*^«3  °iove    forward 

with  a  constant  velocity, 
which  we  shall  call  Y.  The  wavy  form  is  sinusoidal  in  section 
but  of  course  it  is  not  fixed,  but  keeps  moving  ahead.  Hence 
the  moving  sinusoid  of  Fig.  140  may  be  looked  upon  as  a  repre- 
sentation of  this  kind  of  phenomena. 

The  curve  described  on  the  moving  plate  UK  of  Fig.- 140,  if 
referred  to  coordinate  axes  moving  with  the  plate,  is  the  sinusoid,  or 


§187]      SIMPLE  HARMONIC  MOTION  AND  WAVES        346 

sine  curve,  whose  equation  is  (1)  above.  If,  however,  we  consider 
this  curve  as  referred  to  the  fixed  origin  Oi,  then  the  moving 
sinusoid  thus  conceived  is  called  a  simple  progressive  sinusoidal 
wave  or  merely  a  wave.  Under  the  conditions  represented  in 
Fig.  140,  it  is  a  wave  progressing  to  the  right  with  the  uniform 
speed  of  the  plate  HK.  At  any  single  instant,  the  equation  of 
the  curve  is 

y  =  a  sin  h{x  -  OiN),  (2) 

where  OiN  is  the  distance  that  the  node  N  has  been  translated 
to  the  right  of  the  origin  Oi.  If  V  be  the  uniform  velocity  of 
translation  of  HK,  then, 

OiN  =  Vt  (3)1 


Fig.  142. 


and  the  equation  of  the  wave  is 

y  =  asiah{x  —  Vt), 
or  y  =  a  sin  Qix  —  hVt), 

or  y  =  a  sin  (hy  —  kt), 

if  k  be  put  for  hV,  so  that 

V  =  - 


(4) 


(5) 


Because  of  the  presence  of  the  variable  t,  (4)  is  not  the  equation 
of  a  fixed  sinusoid,  but  of  a  moving  sinusoid  or  wave. 

Applying  the  same  terms  used  fbr  S.H.M.,  the  expression 
{hx  —  ht)  is  the  phase  angle,  the  expression  (+  kt)  is  the  epoch 
angle  and  a  is  the  amplitude  of  the  wave.    See  Fig.  143a  and  c. 

The  expression  Qix  —  kt)  is  a  linear  function  of  the  variables 

1  In  what  follows,  t  is  not  the  time  elapsed  since  itf ,  Fig.  140,  was  at  A,  as  used  in 
S183|  but  is  the  elapsed  time  since  N  was  at  0i,  These  values  of  t  differ  by  the  time 
of  half  a  revolution  or  by  ir/k. 


346         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§187 


X  and  t.    The  sine  or  cosine  of  this  function  is  called  a  simple 
harmonic  fmiction  of  x  and  t. 

The  wave  form  on  the  surface  of  water  moves  along  with  fixed 
velocity  V.    The  particles  of  water,  however,  do  not  share  in  this 


b 


^''S<^^'yr> 


<. 


X  X  X/ 


Fig.  143. — Wave  forms,  (a)  of  different  amplitude;  (5)  of  different 
wave  lengths;  (c)  of  different  phase  or  epoch  angles. 

forward  motion.    Each  particle  on  the  surface  moves  up  and 
down  in  a  vertical  line  as  the  wave  form  passes  it.    In  fact  we 
shaU  now  see  that  each  particle  describes  S.H.M.  in  a  vertical 
direction. 
To  examine  the  motion  of  a  single  particle  of  water,  we  have 


§188]      SIMPLE  HARMONIC  MOTION  AND  WAVES         347 

only  to  regard  x  as  constant,  say  x  =  Xi,  in  equation  (4)  above 
The  displacement  of  this  particle  is  then  given  by 

y  =  a  sin  (hxi  —  kt) 
or  y  =  —  a  sin  (M  —  hxi). 

That  is  y  =  a  sin  {kt  —  hxi  —  ir).  (6) 

This  is  the  equation  of  a  S.H.M.  whose  period  is  T  =  2T/k.  The 
epoch  angle  is  hxi  +  ir.  This  will  be  different  for  different  par- 
ticles. This  means  that  the  phase  angles  of  the  S.H.M.  of  succes- 
sive particles  differ,  but  they  all  oscillate  up  and  down  with  the 
same  period  2ir/k. 

188.  Wave  Length.  The  wave  length  of  a  progressive  wave  is 
the  distance  from  crest  to  crest  or  from  trough  to  trough.  It 
is  the  amount  by  which  x  must  be  increased  in  the  equation  of 
the  wave  in  order  that  the  angle  (hx  —  kt)  may  be  increased  by 
2ir.     Hence  the  wave  length, 

^  =  ¥-  « 

189.  Period  or  Periodic  Time.  If  we  fix  our  attention  upon  any 
particular  or  constant  value  of  x,  and  view  the  progressive  wave 
as  it  passes  the  vertical  line  through  this  abscissa,  the  elapsed 
time  from  the  passage  of  one  crest  to  the  next  crest  is  called  the 
period,  or  periodic  time.  It  is  readily  seen  to  be  the  increment 
in  t  which  changes  the  angle  (hx  —  kt)  by  the  amount  2ir.  Hence 
the  period 

The  expression  T  is  called  the  periodic  time,  or  period,  of  the 
wave.  It  is  the  length  of  time  required  for  the  wave  to  move  one 
wave  length.  To  contrast  wave  length  and  period,  think  of  a  per- 
son in  a  boat  anchored  at  a  fixed  point  in  a  lake.  The  time  that 
the  person  must  wait  at  that  fixed  point  (x  constant)  for  crest  to 
follow  crest  is  the  periodic  time.  The  wave  length  is  the  distance 
he  observes  between  crests  at  a  given  instant  of  time  (t  constant) . 
The  number  of  periods  per  unit  of  time  is  called  the  frequency 
of  the  wave.    Hence,  if  N  represent  the  frequency  of  the  wave, 

N  =  |'=|-  (2, 


348        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§190 

190.  Velocity  or  Rate  of  Propagation.  The  rate  of  movement 
V  of  the  sinusoid  on  the  plate  HK,  Fig.  140,  is  shown  by  equation 
(5),  §187,  to  be  k/h  units  of  length  per  second.  This  is  called  the 
velocity  of  the  wave  or  the  velocity  of  propagation.  The  equa- 
tion of  the  wave  may  be  written 

2/  =  o  sin  h{x  —  Yt). 

From  equations  (1)  §188  and  (1)  §189  we  obtain 


A; 
and  since  7  =  r,  we  have 

K 


k_L 

h~  t' 


V  =^.  (1) 

This  equation  is  obvious  from  general  considerations,  for  the 
wave  moves  forward  a  wave  length  L  in  time  T,  hence  the  speed 

of  the  wave  must  be  m' 

191.  L  and  T  Equation  of  a  Wave.  If  we  solve  equations  (1) 
§188  and  (1)  §189  for  h  and  k  respectively,  and  substitute  these 
values  of  h  and  ifc  in  the  equation 

2/  =  a  sin  {hx  —  kt) 
we  obtain 

■\i-a- 

From  this  form  it  is  seen  that  the  argument  of  the  sine  increases 
by  2ir  when  either  x  increases  by  an  amount  L  or  when  t  increases 
by  the  amount  T.  By  use  of  (1),  §190,  the  last  equation  may 
also  be  written 

^-    -'         -  .(2) 


a  sm2^ 


y 

=  asm 

L^^- 

Vt). 

192. 

Phase, 

Epoch, 

Lead. 

Consider  the  two 

waves 

y 

.  I27r\ 
=  a  sm  y-  (a;  — 

Vt) 

y 

=  a  sin 

2ir  . 
j-ix- 

vt- 

E) 

a) 

(2) 
The  amplitudes,  the  wave  lengths  and  the  velocities  are  the 


§192]      SIMPLE  HARMONIC  MOTION  AND  WAVES         349 

same  in  each,  but  the  second  wave  is  in  advance  of  the  first  by 
the  amount  E  (measured  in  linear  units),  for  the  second  equation 
can  be  obtained  from  the  first  by  substituting  (x  —  E)  for  x,  which 
translates  the  curve  the  amount  E  in  the  OX  direction.  In  this 
case  E  is  called  the  lead  (or  the  lag  if  negative)  of  the  second  wave 
compared  with  the  first.  The  lead  is  a  linear  magnitude  measured 
in  centimeters,  inches,  feet,  etc. 

The  terms  phase  and  epoch  are  sometimes  used  to  designate 
the  time,  or,  more  accurately,  the  fractional  amount  of  the  period 
required  to  describe  the  phase  angle  and  epoch  angle  respectively. 
In  this  use,  the  phase  is  the  fractional  part  of  the  period  that  has 
elapsed  since  the  moving  point  last  passed  through  the  middle  point 
of  its  simple  harmonic  motion  in  the  direction  reckoned  as  positive. 
See  Fig.  143c. 

The  tidal  wave  in  mid-ocean,  the  ripples  on  a  water  surface, 
the  wave  sent  along  a  rope  that  is  rapidly  shaken  by  the  hand, 
are  illustrations  of  progressive  waves  of  the  type  discuseed  above. 
Sound  waves  also  belong' to  this  class  if  the  alternate  condensations 
and  rarefactions  of  the  medium  be  graphically  represented  by 
ordinates.  The  ordinary  progressive  waves  observed  upon  a  lake 
or  the  sea  are  not,  however,  progressive  waves  of  this  type.  The 
surface  of  the  water  in  this  case  is  not  sinusoidal  in  form,  but 
is  represented  by  another  class  of  curves  known  in  mathematics 
as  trochoids. 

Exercises 

1.  Derive  the  amplitudOj  the  wave  length,  the  periodic  time,  the 
velocity  of  propagation  of  the  following  waves : 

(a)  y  =  a  sin  {2x  —  3<).  ,  >  .„„    .    2w,         _..        .. 

(b)  y  =5  sin  (0.75a;  -  lOOOi).  W  V  =  10°  ^25^"^  ~  ^°'  "  ^^■ 

(c)  2/  =  10  sin  (I  -  .*)  .  (/)  2/  =  100  sin(5x  +  4t). 

2,r  (?)  y  =  0-025  sin  ^(,x  +  </3). 

id)  y  =  50smy(a;  -  3t).  ■* 

2.  Write  the  equation  of  a  progressive  sinusoidal  wave  whose  height 
is  5  feet,  length  40  feet  and  velocity  4  miles  per  hour. 

3.  Write  the  equation  of  a  wave  of  wave  length  10  meters,  height  1 
meter,  and  velocity  of  propagation  3.5  miles  per  hour.  (Note:  1 
mile  =  1.609  kilometers.) 


350         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§193 

4.  Sound  waves  of  all  wave  lengths  travel  in  still  air  at  70°  F.  with 
a  velocity  of  1130  feet  per  second.  Find  the  wave  length  of  sound 
waves  of  frequencies  256,  128,  and  600  per  second. 

193.  Stationary  Waves.  The  form  of  a  violin  string  during  its 
free  vibration  is  sinusoidal,  but  the  nodes,  crests,  troughs,  etc., 
are  stationary  and  not  progressive  as  in  the  case  of  the  waves 
just  discussed.  Such  waves  are  called  stationary  waves.  The 
water  in  a  basin  or  even  in  a  large  pond  or  lake  is  also  capable  of 
vibrating  in  this  way.  Fig.  144  may  be  used  to  illustrate  the 
stationary  waves  of  this  type,  either  of  a  musical  string  or  of  the 
water  surface  of  &  lake,  but  in  the  case  of  a  vibrating  string,  the 
ends  must  be  supposed  to  be  fastened  at  the  points  0  and  N. 
The  shores  of  the  lake  may  be  taken  at  /  and  K-ot  at  I  and  H, 
etc.    As  is  well  known,  such  bodies  are  capable  of  vibrating  in 


Fig.  144. — A  stationary  wave. 


segments  so  that  the  number  of  nodes  may  be  large.  This 
explains  the  "harmonics"  of  a  vibrating  violin  string  and  the 
various  modes  in  which  stationary  waves  may  exist  on  a  water 
surface.  A  stationary  wave  on  the  surface  of  a  lake  or  pond  is 
known  as  a  seiche,  and  was  first  noted  and  studied  on  Lake 
Geneva,  Switzerland.  The  amplitudes  of  seiches  are  usually 
small,  and  must  be  studied  by  means  of  recording  instruments 
so  set  up  that  the  influence  of  progressive  waves  is  eliminated. 
The  maximum  seiche  recorded  on  Lake  Geneva  was  about  6  feet, 
although  the  ordinary  amplitude  is  only  a  few  centimeters. 

The  equation  of  a  stationary  wave  may  be  found  by  adding  the 
ordinates  of  a  progressive  wave 

y  =  asin  (hx  —  kt)  (1) 


§193]      SIMPLE  HARMONIC  MOTION  AND  WAVES         351 

traveling  to  the  right  (A;  >  0),  to  the  ordinates  of  a  progressive 
wave 

y  =  asm  {hx  +  /c<)  (2) 

traveling  to  the  left. 

Expanding  the  right  members  of  (1)  and  (2)  by  the  addition 
formula  for  the  sine  and  adding 

y  =  2a  cos  kt  sin  hx,  (3) 

or  in  terms  of  L  and  T,  §188  (1)  and  §189  (1), 

'   y  =  2a cos  (^)  sin  {~:j  ■  (4) 

In  Fig.  144,  the  origin  is  at  0  and  the  X-axis  is  the  Line  of  nodes 
ONX.  If  in  equation  (3)  we  look  upon  2a  cos  kt  as  the  vari- 
able amplitude  of  the  sinusoid 

y  =  sin  hx, 

we  note  that  the  nodes,  of  the  sinusoid  remain  stationary,  but 
that  the  amplitude  2a  cos  kt  changes  as  time  goes  on.  When 
t  =  0,  the  sine  curve  has  amplitude  2a  and  wave  length  2ir/h. 
When  t  =  ir/2k,  or  T/i,  the  sinusoid  is  reduced  to  the  straight 
line  y  =  0.     When  t  =  ir/k,  or  T/2,  the  curve  is  the  sinusoid 

y  =  —  2a  sin  hx 

which  has  a  trough  where  the  initial  form  had  a  crest,  or  vice 
versa. 

Exercises 

In  the  following  exercises  the  height  of  the  wave  means  the  maxi- 
mum rise  above  the  line  of  nodes.  When  a  seiche  is  uninodal,  the 
shores  of  the  lake  correspond  to  the  points  I  and  K,  Fig.  144.  When 
a  seiche  is  binodal,  the  points  /  and  H  are  at  the  lake  shore. 

1.  From  the  equation  of  a  stationary  wave  in  the  form  y  = 
2a  sin  %rx/L  cos  2-wtlT,  show  that  K,  Fig.  144,  is  at  its  lowest  depth 
fori  =  r/2,,3r/2,  67/2,    . 

2.  Henry  observed  a  fifteen-hour  uninodal  seiche  in  Lake  Erie, 
which  was  396  kilometers  in  length.  Write  the  equation  of  the  prin- 
cipal or  uninodal  stationary  wave  if  the  amplitude  of  the  seiche  was 
15  cm. 

3.  A  small  pond  111  meters  in  length  was  observed  by  Eridros  to 
have  a  uninodal  seiche  of  period  fourteen  seconds.  Write  the  equation 
of  the  stationary  wave  if  the  ampUtude  be  o. 


352        ELEMENTARY  MATHEMATICAL  ANALYSIS     [§194 

4.  Forel  reports  that  the  uninodal  longitudinal  seiche  of  Lake 
Geneva  has  a  period  of  seventy-three  minutes  and  that  the  binodal 
seiche  has  a  period  of  thirty-five  and  one-half  minutes.  The  trans- 
verse seiche  has  a  period  of  ten  minutes  for  the  uninodal  and  five 
minutes  for  the  binodal.  The  longitudinal  and  transverse  axes  of  the 
lake  are  45  miles  and  5  miles  respectively.  Write  the  equation  of 
these  different  seiches. 

5.  A  standing  wave  or  uninodal  seiche  exists  on  Lake  Mendota  of 
period  twenty-two  minutes.  If  the  maximum  height  is  8  inches  and 
the  distance  .across  the  lake  is  6  miles,  write  the  equation  of  the  seiche. 

194.*  Compound  Harmonic  Motion  and  Compound  Waves. 
The  addition  of  two  or  more  simple  harmonic  functions  of  frequencies 
which  are  multiples  of  the  frequency  of  a  given  first  or  fundamental 
harmonic,  gives  rise  to  compound  harmonic  motion.     Thus, 

y  =  a  sin  fc<  +  &  sin  Zkt, 

corresponds  to  the  superposition  of  a  S.H.M.  of  period  2ir/3fc  and 
amplitude  6  upon  a  fundamental  S.H.M.  of  period  2ir/A;  and  amplitude 
a.  To  compound  motions  of  this  type,  there  correspond  compound 
waves  of  various  sorts,  such  as  a  fundamental  sound  wave  with 
overtones,  or  tidal  waves  in  restricted  bays  or  harbors.  The  graphs 
of  the  curves 

y  =  sin  X  +  sin  2x 

y  =  ainx  +  sin  3a 

are  easily  constructed.  They  may  be  drawn  by  adding  the  ordinates 
of  the  various  sinusoids  constructed  on  the  same  axis,  as  in  Fig.  145. 
To  compound  the  curves,  first  draw  the  component  curves,  say  y  = 
sin  X  and  y  =  sin  3x  of  Kg.  145.  Then  use  the  edge  of  a  piece  of  paper 
divided  proportionally  to  sin  x  (that  is,  like  the  scale  OB,  Fig.  145)  and 
use  this  as  a  scale  by  means  of  which  the  successive  ordinates  of  a  given 
X  may  be  added.  For  example,  to  locate  the  point  on  the  composite 
curve  corresponding  to  the  abscissa  OD,  Fig.  145,  we  must  add  DP 
and  DQ.  Hence  place  vertically  at  P  the  lower  end  of  the  paper  scale 
just  mentioned.  The  sixth  scale  division  above  P  on  this  scale  will 
then  locate  the  required  point  M  of  the  composite  scale.' 
In  Fig.  146  the  curves: 

y  =  sin  X  +  sin  (2x  +  27rn/16) 

y  =  sin  2x  +  sin  (3x  +  2)rre/16) 

I  Note  that  if  the  method  described  be  used,  there  is  really  no  need  of  drawing 
the  curve  y  =  sin  3a:.  If  both  curves  are  drawn,  ordinates  may  conveniently  be 
added  with  bow  dividers. 


§194]      SIMPLE  HARMONIC  MOTION  AND  WAVES         353 


are  shown  for  values  of  n  =  0,  1,  2,  .  ,  IS  in  succession — that  is, 

for  successive  phase  differences  corresponding  to  one-sixteenth  of  the 
wave  length  of  the  fundamental  y  =  sin  x. 


Fig.  145. — The  curves  y  =  sin  x,y=  sin  3x  and  the  compound  curve 
y  =  sin  X  +  sin  3x.  '^ 


Fifth 


Fig.  146. — The  curves  (o)  j/  =  sin  x  +  sin  {2x  +  27rn/16)  and  (b) 
2/  =  sin  2a;  +  sm  (3x  +  2Tn/16),  for  n  =  0,  1,  2,  .  .  15.  {From 
Thomson  and  Tail.) 

Wave  forms  compounded  from  the  odd  harmonics  only  are  espe- 
eially  important,  as  alternating-current  curves  are  of  this  type.  See 
Fig.  147. 

23 


354        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§196 

196.*  Harmonic  Analysis.  Fourier  showed  in  1822  in  his  "Ana- 
lytical Theory  of  Heat"  that  a  periodic  single-valued  function,  say 
y  —  f(x),  under  certain  conditions  of  continuity,  can  be  represented 
by  the  sum  of  a  series  of  sines  and  cosines  of  the  multiple  angles  of  the 
form 

y  =  ao  +  ai  cos  x  +  a^  cos  2x  +  Oa  cos  3x  +  .    .    . 
+  bi  sin  X  +bi  sin  2a;  -|-  63  sin  3a;  +  .    .    . 

This  means,  for  example,  that  it  is  always  possible  to  represent  the 
complex  tidal  wave  in  a  harbor,  by  means  of  the  sum  of  a  number  of 
simple  waves  or  harmonics.  The  term  harmonic  analysis  is  given  to 
the  process  of  determining  these  sinusoidal  components  of  a  compound 
periodic  curve.     In  §194  we  have  performed  the  direct  operation  of 


50 

V 

/ 

/ 

\ 

26 

/ 

\ 

/ 

s 

/ 

\ 

S 

0 

/ 

■     1 

)    1 

2     1 

1  1 

6H 

)    ! 

2    2 

4    2 

i    2 

i      £ 

0    3 

!     3 

V 

\ 

> 

25 

\ 

/. 

V 

/ 

60 

\ 

/ 

~" 

' 

Fig.  147. — An  alternating  current  curve. 

present. 


Only  odd  harmonics  are 


finding  the  compound  curve  when  the  component  harmonics  are  given. 
The  inverse  operation  of  finding  the  components  when  the  compound 
curve  is  given  is  much  more  difficult,  and  its  discussion  must  be  post- 
poned to  a  later  course. 

196.*  Connecting  Rod  Motion.  If  one  end  of  a  straight  Une  B  be 
required  to  move  on  a  circle  while  the  other  end  of  the  line  A  moves  on 
a  straight  Une  passing  through  the  center  of  the  circle,  the  resulting 
motion  is  Icnown  as  connecting  rod  motion.  The  connecting  rod  of  a 
steam  engine  has  this  motion,  as  the  end  attached  to  the  crank  travels 
in  a  circle  while  the  end  attached  to  the  cross-head  travels  in  a 
straight  line.  The  motion  of  the  end  A,  Fig.  148,  of  the  connecting 
rod  is  approximately  S.H.M.     The  approximation  is  very  close  if  the 


§196]      SIMPLE  HARMONIC  MOTION  AND  WAVES         355 


connecting  rod  be  very  long  in  comparison  with  the  diameter  of  the 
circle. 

A  second  approximation  to  the  motion  of  the  point  A  can  be 
obtained  by  introducing  the  second  harmonic  or  octave  of  the  funda- 
mental. In  Fig.  148,  let  the  radius  of  the  circle  be  a  and  the  length 
of  the  connecting  rod  be  I.  The  length  of  the  stroke  M'N  is  2a,  and 
the  origin  may  conveniently  be  taken  at  the  mid-point  of  the  stroke, 
0.  When  B  is  at  E,  A  is  at  M  and  when  B  is  at  K,  A  is  at  A'^. 
Then  MH  =  NK  =  I  and  OC  =  I.     Now 


But 
and 
Hence 


X  =  CA  -  CO  =  CA  -  I  =  CD  +  DA  -  I. 
CD  =  a  cos  e 
DA  =  Vl^  -  BD'  =  Vl'  -  a^  sin^  e. 
X  =  acose  +1  Vl  -  (a^/l^)  sin^  B  -  I 


(1) 
(2) 
(3) 
(4) 


Fig.  148. — Connecting  rod  motion. 


Approximating  the  radical  by  §113  (\/l  —  x  =  1  —  x/2)  we  obtain 

^    ,   ,  / ,       a^  sin^  e\        ,  ,,, 

X  =  acos  9  +1  il 2p — )   ~  '•  (^) 

Since  sin^  9  =  (1  —  cos  26) /2,  we  obtain 

X  =  a  cos  9+27  "^"^  ^^  ~  47'  ^^^ 

which  is  approximately  true  as  long  as  I  is  much  greater  than  a. 

It  is  seen  from  the  above  result  that  the  second  approximation  to 
connecting  rod  motion  contains  as  overtone  the  octave,  or  second 

a* 
harmonic,  ^j  cos  29,  in  addition  to  the  first  or  fundamental  harmonic 

a  cos  8. 


356        ELEMENTARY  MATHEMATICAL  ANALYSIS      [196  § 

Exercises 

1.  Draw  the  curve  corresponding  to  equation  (5)  above  if  o'  =  1.15 
inches,  and  Z  =  3  inches. 

2.  The  motion  of  a  slide  valve  is  given  by  an  equation  of  the  form 

3/  =  oi  sin  (9  +  e)  +  02  sin  (28  +  90°). 

Draw  the  curve  if  ai  =  100,  oj  =  25,  c  =  40°,  using  6  as  the  abscissa 
in  rectangular  coordinates. 


CHAPTER  XII 
COMPLEX  NUMBERS 

197.  ScaJes  of  Numbers.  To  measure  any  magnitude,  we 
apply  a  unit  of  measure  and  then  express  the  result  in  terms  of 
numbers.  Thus,  to  measure  the  volume  of  the  liquid  in  a  cask 
we  may  draw  off  the  liquid,  a  measure  full  at  a  time,  in  a  gallon 
measure,  and  conclude,  for  example,  that  the  number  of  gallons 
is  125.  In  this  case  the  number  12^  is  taken  from  the  arith- 
metical scale  of  numbers,  0,  1,  2,  3,  4,  .  .  If  we  desire  to  meas- 
ure the  height  of  a  stake  above  the  ground,  we  may  apply  a  foot- 
rule  and  say,  for  example,  that  the  height  in  inches  above  the 
ground  is  12|,  or,  if  the  positive  sign  indicates  height  above  the 
ground,  we  may  say  that  the  height  in  inches  is  -|-  12J.  In 
this  latter  case  the  number  -|-  12?  has  been  selected  from  the 
algebraic  scale  of  numbers  .  .  .  —  4,  —  3,  —  2,  —  1,  0,  +  1, 
+  2,  +  3,  +  4,       .    . 

The  scale  of  numbers  which  must  be  used  to  express  the  value  of  a 
magnitude  depends  entirely  upon  the  nature  of  the  magnitude.  The 
attempt  to  express  certain  magnitudes  by  means  of  numbers  taken 
from  the  algebraic  scale  may  sometimes  lead,  as  every  student  of 
algebra  knows,  to  meaningless  absurdities.  Thus  a  problem  involving 
the  number  of  sheep  in  a  pen,  or  the  number  of  marbles  in  a  box,  or 
the  number  of  gallons  in  a  cask,  cannot  lead  to  a  negative  result,  for 
the  magnitudes  just  named  are  arithmetical  quantities  and  their  meas- 
urement leads  to  a  number  taken  from  the  arithmetical  scale.  The 
absurdity  that  sometimes  appears  in  results  to  problems  concerning 
these  magnitudes  is  due  to  the  fact  that  one  attempts  to  apply  the 
notion  of  algebraic  number  to  a  magnitude  that  does  not  permit  of  it. 
Science  deals  with  a  great  many  different  kinds  of  magnitudes,  the 
measurement  of  some  of  which  leads  to  arithmetical  numbers  while  the 
measurement  of  others  leads  to  algebraic  numbers;  the  remarkable 
fact  is  that  two  different  number  scales  serve  adequately  to  express 
magnitudes  of  so  many  different  sorts.'    The  magnitudes  of  science 

357 


358        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§198 

are  so  various  in  kind  that  one  might  reasonably  expect  that  the 
variety  of  number  systems  required  in  the  mathematics  of  these 
sciences  would  be  very  great. 

The  arithmetical  scale  is  used  when  we  enumerate  the  number  of 
gallons  in  a  cask  and  say:  0,  1,  2,  3,  .  .  .  If  we  observe  3  gallons 
in  the  cask,  and  then  remove  one,  we  note  those  remaining  and  say 
tiDo;  we  may  remove  another  gallon  and  say  one,  we  may  remove  the 
last  gallon  and  say  zero;  but  now  the  magnitude  has  come  to  an  end. 
The  algebraic  scale  is  used  when  we  measure  in  inches  the  height  of 
a  stake  above  the  ground  and  say  three.  We  may  drive  the  stake 
down  an  inch  and  say  two;  we  may  drive  the  stake  another  inch  and 
say  one;  we  may  drive  the  stake  another  inch  and  say  zero,  or 
"level  with  the  ground;"  but,  unUke  the  case  of  the  gallons  in  the 
cask,  we  need  not  stop  but  may  drive  the  stake  another  inch  and  say 
one  below  the  ground,  or,  for  brevity,  minus  one;  and  so  on. 

Many  of  the  magnitudes  considered  in  science  are  completely  ex- 
pressed by  means  of  arithmetical  numbers  only;  for  example,  such 
magnitudes  as  density  or  specific  gravity;  temperature;^  electrical  re- 
sistance; quantity  of  energy;  such  as  ergs,  joules  or  foot-pounds; 
power,  such  as  horse  power,  kilowatts,  etc.  All  of  the  magnitudes 
just  mentioned  are  scalar,  as  it  is  called;  that  is,  they  exist  in  one 
sense  only — ^not  in  one  sense  and  also  in  the  opposite  sense,  as  do  forces, 
velocitiesj  distances,  as  explained  above.  The  arithmetical  scale  of 
numbers  is  therefore  ample  for  their  expression. 

The  distraction,  then,  between  an  algebraic  number  and  an  arith- 
metical number  is  the  notion  of  sense  which  must  always  be  associated 
with  any  algebraic  number.  Thus  an  algebraic  number  not  only 
answers  the  question  "how  many"  but  also  affirms  the  sense  in  which 
that  number  is  to  be  understood;  thus  the  algebraic  number  -|-  12 J,  if 
arising  in  the  measurement  of  angular  magnitude,  refers  to  an  angular 
magnitude  of  12|  units  (degrees,  or  radians,  etc.)  taken  in  the  sense 
defined  as  positive  rotation. 

198.  Algebraic  Number  Not  the  Most  General  Sort.  Algebraic 
numbers,  although  more  general  than  arithmetical  numbers,  are 
themselves  quite  restricted.  For,  each  algebraic  number  corre- 
sponds to  a  point  of  the  algebraic  scale.  But  for  points  not  on  the 
scale  there  corresponds  no  algebraic  number.  That  is,  the  alge- 
braic scale  is  one-dimensional.    It  is  thus  seen  that  there  is  an 

^  Temperature  is  an  arithmetical  quantity,  since  there  is  an  absolute  zero  of 
temperature.  Temperature  does  not  exist  in  two  opposite  senses,  but  in  a  single 
sense. 


§199]  COMPLEX  NUMBERS  359 

opportunity  of  enlarging  our  conception  of  number  if  we  can  re- 
move the  restriction  of  one  dimension — that  is,  if  we  can  get  out  of 
the  line  of  the  algebraic  scale  and  set  up  a  number  system  such  that 
one  number  of  the  system  will  correspond,  for  examiple,  to  each 
point  of  a  plane,  and  such  that  one  point  of  the  plane  will  corre- 
spond to  each  number  of  the  system.  We  will  seek,  therefore,  an 
extension  or  generalization  of  the  number  system  of  algebra  that 
will  enable  us  to  consider,  along  with  the  points  of  the  algebraic 
scale,  those  points  which  lie  without  it. 

199.  Numbers  as  Operators.  The  extension  of  the  number 
system  mentioned  in  the  last  section  may  be  facilitated  by  chang- 
ing the  conception  usually  associated  with  symbols  of  number. 
The  usual  distinction  in  algebra  is  between  symbols  of  number  and 
symbols  of  operation.  Thus  a  symbol  which  may  be  looked  upon  as 
answering  the  question  "how  many"  is  called  a  number,  whUe  a 
symbol  which  tells  us  to  do  something  is  called  a  symbol  of  opera- 
tion, or,  simply,  an  operator.  Thus  in  the  expression  -\/2)  "v/  is 
a  symbol  of  operation  and  2  is  a  number.  A  symbol  of  operation 
may  always  be  read  as  a  verb  in  the  imperative  mood;  thus  we 
may  read  -s/x:  "Take  the  square  root  of  x."  Likewise  log  x, 
and  cos  9  may  be  read;  "Find  the  logarithm  of  x,"  "Take  the  co- 
sine of  8."  In  these  expressions  "log"  and  "cos"  are  symbols 
of  operation;  they  teU  us  to  do  something;  they  do  not  answer  the 
question  "how  many"  or  "how  much"  and  hence  are  not  num- 
bers. Here  we  speak  of  -\/j  log,  cos,  as  operators ;  we  speak  of  x  as 
.  the  operand,  or  that  which  is  operated  upon. 

It  is  interesting  to  note  that  any  number  may  be  regarded  as  a 
symbol  of  operation;  by  doing  so  we  very  greatly  enlarge  some 
original  conceptions.  Thus,  10  may  be  regarded  not  only  as  ten, 
answering  the  question  "how  many,"  but  it  may  quite  as  well  be 
regarded  as  denoting  the  operation  of  taking  unity,  or  any  other 
operand  that  follows"  it,  ten  times;  to  express  this  we  may  write 
10-1,  in  which  10  may  be  called  a  tensor  (that  is,  "stretcher"), 
or  a  symbol  of  the  operation  of  stretching  a  unit  until  the  result 
obtained  is  tenfold  the  size  of  the  unit  itself.  In  the  same  way 
the  symbol  2  may  be  looked  upon  as  denoting  the  operation  of 
doubling  unity,  or  of  doubling  any  operand  that  follows  it;  like- 


360        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§199 

wise  the  tensor  3  may  be  looked  upon  as  a  trebler,  4  as  a  quadrupler, 
etc. 

With  the  usual  understanding  that  any  symbol  of  operation 
operates  upon  that  which  follows  it,  we  may  write  compound 
operators  like  2-2-3-1.  Here  3  denotes  a  trebler  and  31  denotes 
that  the  unit  is  to  be  trebled,  2  denotes  that  this  result  is  to  be 
doubled  and  the  next  2  denotes  that  this  result  is  to  be  doubled. 
Thus  representing  the  unit  by  a  line  running  to  the  right,  we  have 
the  following  representation  of  the  operators : 

The  unit  -> 

3-1  -^^-^ 

2-31 > > 

2-2-31  T > > 


Notice  the  significance  that  should  now  be  assigned  to  an  expo- 
nent attached  to  these  (or  other)  symbols  of  operation.  The 
exponent  means  to  repeat  the  operation  designated  by  the  operator; 
that  is,  the  operation  designated  by  the  base  is  to  be  performed, 
and  performed  again  on  this  result,  and  so  on,  the  number  of  opera- 
tions being  denoted  by  the  exponent.  Thus  W  means  to  perform 
the  operation  of  repeating  unity  ten  times  (indicated  by  10)  and 
then  to  perform  the  operation  of  repeating  the  result  ten  times, 
that  is,  it  means  10  (101).  Also,  10'  means  10[10(10-1)].  Like- 
wise log^  30  means  log  (log  30)  which,  if  the  base  be  10,  equals 
log  1.4771,  or  finally  0.1694.  An  apparent  exception- occurs  in 
the  case  of  the  trigonometric  functions.  The  expression  cos'j; 
should  mean,  in  this  notation,  cos  (cos  x),  but  because  trigo- 
nometry is  historically  so  much  older  than  the  ideas  here  ex- 
pressed, the  expression  cos''  x  came  to  be  used  for  (cos  a;)',  or 
(cos  x)  X  (cos  x),  but  cos~^  6  means  arc  cos  6,  not  1/cos  9. 

To  be  consistent  with  the  notation  of  elementary  mathematics, 
the  expression  \/4,  looked  upon  as  a  symbol  of  operation,  must 
denote  an  operation  which  must  be  performed  twice  in  order 
to  be  equivalent  to  the  operation  of  quadrupling;  that  is,  such 
that  (-\/4)^  =  4.  Likewise  i/i  denotes  an  operation  which 
must  be'  performed  three  times  in  succession  in  order  to  be 
equivalent  to  quadrupling.  But  we  know  that  the  operation 
denoted  by  2,  if  performed  twice,  is  equivalent  to  quadrupling; 


§200]  COMPLEX  NUMBERS  361 

therefore  \/4  =  2,  etc.  Just  as  4^,  4',  etc.,  may  be  called  stronger 
tensors  than  a  single  4,  so  -s/i,  Vi,  etc.  may  be  called  weaker 
tensors  than  the  operator  4. 

200.  Reversor.  The  expression  (  —  1),  looked  upon  as  a 
symbol  of  operation,  is  not  a  tensor,  as  it  leaves  the  size  unchanged 
of  that  upon  which  it  operates.  If  this  operator  be  applied  to 
any  magnitude,  it  will  change  the  sense  in  which  the  magnitude 
is  then  taken  to  exactly  the  opposite  sense.  Thus,  if  6  stands 
for  six  hours  after,  then  (  —  1)(6)  stands  for  six  hours  before 
a  certain  event,  and  (  —  1)  is  the  sj'mbol  of  this  operation  of 
reversing  the  sense  of  the  magnitude.  Also  if  6  stands  for  a  line 
running  six  units  to  the  right  of  a  certain  point,  then  (  —  1)(6) 
stands  for  a  line  running  six  units  to  the  left  of  that  point;  so 
that  (  —  1)  is  the  symbol  which  denotes  the  operation  of  turning 
the  straight  line  through  180°.  As  2,  3,  4,  when  looked  upon  as 
symbols  of  operations,  were  called  tensors,  the  operator  (  —  1) 
may  conveniently  be  designated  a  reversor. 

Exercises 

Show  graphically  the  effect  of  the  operations  indicated  in  each  of 
the  following  exercises.  Take  as  the  initial  unit-operand  a  straight 
line  1/2  inch  long  extending  to  the  right  of  the  zero  or  initial  point. 
Explaia  each  expression  as  consisting  of  the  operand  unity  and 
symbols  of  operation — ^tensors,  reversors,  etc.,  which  operate  upon 
it,  one  after  the  other,  in  a  definite  order. 

1.  2-3-1.  8.  (Viy-i  - 1)-1- 

2.  3-3-1.  9.  (  -  l)s-22-31. 

3.  -  1-31.  10.  3-321. 

4.  2'1.  11.  (  -  1)'2-2«1. 

6.  VSI.  12.  3(  -  1)V21. 

6.  (-v/2)^-l.  13.  (\/2)-(  -  1)"»-1. 

7.  -v/gVi-l.  14.    Vl0-2-(  -  1)1. 

15.  A  tensor,  if  permitted  to  operate  seven  times  in  succession,  will 
just  double  the  operand.     Symbolize  this  tensor. 

16.  A  tensor,  if  permitted  to  operate  five  times  in  succession,  will 
quadruple  the  operand.     Symbohze  this  tensor. 


362        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§201 


201.  Versors.  The  expression  ■%/  —  1  cannot  consistently, 
with  the  meaning  abeady  assigned  to  \/  and  (  —  1),  be  looked 
upon  as  answering  the  question  "how  many,"  and  therefore  is  not 
a  number  in  that  sense;  yet  if  we  consider  \/  —  1  as  a  symbol  of 
operation,  it  can  be  given  a  meaning  consistent  with  the  operators 
already  considered.  For  if  2  is  the  operator  that  doubles,  and 
\/2  is  the  operator  that  when  used  twice  doubles,  then  since  (  —  1) 
is  the  operator  that  reverses,  the  expression  \/  —  ^  should  be  an 
operator  which,  when  used  twice,  reverses.  So,  as  (  —  1)  may 
be  defined  as  the  symbol  which  operates  to  turn  a  straight  line 
through  an  angle  of  180°,  in  a  similar  way  we  may  define  the 

expression  ■%/  —  1  as  «  symbol 
which  denotes  the  operation  of 
turning  a  straight  line  through 
an  angle  of  90°  in  the  positive 
direction.  The  restriction  of 
positive  rotation  is  inserted 
to  make  the  definition  unique. 
The  symbols  (  —  1)  and 
■y/  —  1  are  not  tensors.  They 
do  not  represent  a  stretching 
or  contracting  of  the  operand. 
Their  effect  is  merely  to  turn 
the  operand  to  a  new  direc- 
tion; hence  these  symbols 
may  be  called  versors,  or 
"turners." 

202.  The  Operator  V^^.  In  Fig.  149  let  a  be  any  line. 
Then  a  operated  upon  by  V  -  1  (that  is,  V  —  1  a)  is  a  turned 
anti-clockwise  through  90°,  which  gives  OB.  Now,  of  course, 
V—  1  can  operate  on  V  —  1  a  just  as  well  as  on  a.  Then 
V  —  1  ( V  —  1  a),  or  PC,  is  V  —  1  g  turned  positively  through 
90°.  Similarly,  V  -  UV  -  1(V  -  1  a)]  is  V^I  i.s/'^l  a) 
turned  through  90°,  etc. 

As  we  are  at  liberty  to  consider  two  turns  of  90°  as  equivalent 
to  one  turn  of  180°,  therefore,  \/  —  1  (V  —  1  a)  =  (  —  1)  o. 
Now  OD  =  (  -  1)  OS,   OD  =  i-  1)  (-v/^T  a);  but  also  0D  = 


B 

(\Rfa 

e 

0                a            J 

C 

9 

J 

'd 

Fig.   149.- 


-The  integral  powers  of 


§203]  COMPLEX  NUMBERS  363 

V^  (  -  a),  therefore,  (  -  1)  V^  a  =  V"^  (  -  «)•    Thus 
the  student  may  show  many  like  relations. 

The  operator  •%/  —  1  is  usually  represented  by  the  symbol  i  and 
will  generally  be  so  represented  in  what  follows. 


Exercises 

Interpret  each  of  the  following  expressions  as  a  symbol  of  operation: 

1.  2,  3,  4,  -1. 

■2    3^23,4",  (-1^  (-1)^    

3.  V2,VZ,V-  1,  \/'2,  \/-  1. 

Select  a  convenient  unit  and  construct  each  of  the  following  expres- 
sions geometrically,  explaining  the  meaning  of  each  operator: 

4.  2-3-5-1.  7.  (-1)''V^^-1. 

6.  2=-(-l)-l.  8.  2'-(-l)^-(\/-  l)"-!. 

6.  3V-1-21.  9.  3V  -  1(-1)V -11. 

203.  Complex  Numbers.  An  expression  of  the  form  a  +  hi 
is  cdlled  a  complex  number,  since  it  contains  a  term  taken  from 
each  of  the  following  scales,  so  th.at  the  unit  is  not  single  but 
double  or  complex: 

-  3,  ■-  2,  -  1,  0,  +  1,  +  2,  +  3, 
.    -  3i,  -  2i,  -  i,  0,  +  I,  +  2i,  +  3t, 

Any  number  belonging  to  the  first  scale  is  called  a  real  nimiber, 
any  number  belonging  to  the  second  scale  is  called  a  pure 
imaginary. 

It  is  important  to  note  that  the  only  element  common  to  the  two 
series  in  this  complex  scale  is  0. 

The  explanation  of  the  meaning  of  the  symbol  (a  +  hi)  will 
be  given  in  the  following  section.  It  will  be  shown  in  subsequent 
theorems  that  any  expression  made  up  of  the  sum,  product, 
power  or  quotient  of  complex  numbers  may  be  put  in  the  form 
a  +  hi,  in  which  both  a  and  6  are  re&l. 

204.  Meaning  of  a  Complex  Nimiber.  Any  real  number,  or 
any  expression  containing  only  real  numbers,  may  be  consid- 
ered as  locating  a  point  in  a  line. 

Thus,  suppose  we  wish  to  draw  the  expression  2  +  5.    Let  0  be 


364        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§204 


the  zero  point  and  OX  the  positive  direction.  Lay  off  OA  =  2  in 
the  direction  OX  and  at  A  lay  off  AB  =  5  in  the  direction  OX. 
Then  the  path  OA  +  AB  is  the  geometrical  representation  of 
2+5. 

0  A  B  X 


Any  complex  number  may  be  taken  as  the  representation  of  the 
position  of  a  point  in  a  plane.  For,  suppose  c  +  di  is  the  complex 
number.  Let  0,  Fig.  150,  be  the  zero  point  and  OX  the  positive 
direction.    Lay  off  OA  =  +  c  in  the  direction  OX  and  at  A  erect 

di  in  the  direction  OY,  in- 
stead of  in  the  direction  OX 
as  in  the  last  example.  It 
is  agreed  to  consider  the 
step  to  the  right,  OA, 
followed  by  the  step  up- 
ward, AP,  as  the  meaning 
of  the  complex  number  c  + 
di^  Either  the  broken  'path 
OA  +  AP  or  the  direct  -path 
OP  may  he  taken  as  the  repre- 
smtation  of  c  +  di,  and  either 
path  constitutes  the  definition 
of  the  sum  of  c  and  di. 
—  di,  and  —  c  +  di  may  be 


Fig.  150. — The  geometrical  con- 
struction of  a  complex  number, 
c  +  di. 


di. 


In  the  same  manner  c 
constructed. 

The  meaning  of  some  of  the  laws  of  algebra  as  applied  to  imagi- 
naries  may  now  be  illustrated.    Let  us  construct  c  +  di  +  a  +  hi. 

The  first  two  terms,  c  +  di,  give  OA  +  AB,  locating  B  (Fig. 
151).  The  next  two  terms,  a  +  hi,  give  BC  +  CP,  locating  P. 
Hence  the  entire  expression  locates  the  point  P  with  reference  to 
0.  Now  if  the  original  expression  be  changed  in  any  manner 
allowed  by  the  laws  of  algebra,  the  result  is  merely  a  different  path 
to  the  same  point.    Thus: 

c  +  a  +di  +  hiis  the  path  OA,  AD,  DC,  CP 
{c+a)+  {d  +  h)i  is  the  path  OD,  DP 
a  +  di+  c  +  6i  is  the  path  OE,  EH,  HC,  CP 
a+di  +  hi  +  c  is  the  path  OE,  EH,  HF,  FP,  etc. 


§205] 


COMPLEX  NUMBERS 


365 


The  student  should  consider  other  cases.  Is  there  any  method 
of  locating  P  with  the  same  four  elements,  which  the  figure  does 
not  illustrate? 

205.  Laws.  It  can  be  shown  by  simple  geometrical  construc- 
tion that  the  operator  i,  as  defined  above,  obeys  the  ordinary 
laws  of  algebra.  We  can  then  apply  all  of  the  elementary  laws  of 
algebra  to  the  symbol  i  and  work  with  it  just  as  we  do  with  any 
other  letter.    The  following  are  illustrations  of  each  law: 


r      a 

c 

y 

^ ^ 

^ 

. 

1 

F 

G 

f 

•* 

^-N 

•a 

I 

H 

B 

a 

c 

ts 

•■s 

^ 

0 

E 

^ 

, 

. 

A 

D 

Fig.  151. — Illustration  of  the  application  of  the  laws  of -algebra  and 
the  expression  c  +  di  +  a  +  bi. 

CoMMUTATrvE  Law: 

c-\-di-\-a  +  bi  =  c  +  a  +  di-\-bi  =  di-{-c  +  bi  +  a,  etc. 
ai  =  ia,  iai  =  iia  =  aii,  etc. 

Thus  the  equation  lO-s/  —  1 


/  -  llO.or  better,  lOV  -  1-1 

\/  —  l-lOl  may  be  said  to  mean  that  the  result  of  performing 
the  operation  of  turning  unity  through  90°  and  performing  upon  , 
the  result  the  operation  of  taking  it  ten  times,  is  the  same  as  the 
result  of  performing  the  operation  of  taking  unity  ten  times  and 
performing  upon  this  result  the  operation  of  turning  through  90°. 

AssociATivB  Law: 

(c  +  di)  +  (a  +  bi)  =  c  +  {di  +  a)  +  bi,  etc. 
{ab)i  =  a{bi)  =  abi,  etc. 
DiSTEiBUTrvE  Law: 


(a  +  b)i  =  ai  + 


etc. 


366        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§206 

The  expression  -\/  —  a,  where  a  is  any  number  of  the  arith- 
metical  scale,  is  defined  as  equivalent  to  \/  —  l-o;that  is,  y/  —  a 
-  i\fa.  For  example,  V  —  4  =  2i,  V  —3  =  i'\/^,  etc.  In 
what  foUows  it  is  presupposed  that  the  student  will  reduce  expressions 
of  the  form  -y/  —  ato  the  form  i  s/a  before  performing  algebraic  op- 
erations. From  this  it  follows  that  y/  —  a-^/  —  b  =  —  y/cA  and 
not  Vobl 

The  relation  -\/  —  4  =  2-\/  —  1  may  be  interpreted  as  follows : 
(  —  4)  is  the  operator  that  quadruples  and  reverses;  then  •%/  —  4 
is  an  operator  which  used  twice  quadruples  and  reverses.  But 
2-%/  —  1  is  an  operator  such  that  two  such  operators  quadruple 
and  reverse.    That  is,  V  —  4  =  2\/  —  1. 

206.  Powers  of  i.  We  shall  now  interpret  the  powers  of  i  by 
means  of  the  new  significance  of  an  exponent  and  by  the  commu- 
tative, associative  and  other  laws.    First: 

i°  or  i°  1     =  +  1  i^  =  iH    =       i 

^  i'  .or  i^  1     =       i  %'•  =  iH    =  —  \ 

j2  =  _  ]^  j7  ~=  m  =  —  i 

i^  =  iH       =  —  i  i'  =  m    =  +  1 

i*  =  m^      =  +  1  etc.  etc. 

Whence  it  is  seen  that  all  even  powers  of  i  are  either  +  1  or  —  1, 
and  all  odd  powers  are  either  i  or  —  i.  The  student  may  reconcile 
this  with  Fig.  149.  The  zero  power  of  i  must  be  unity,  for  the 
exponent  zero  can  only  mean  that  the  operation  denoted  by  the 
symbol  of  operation  is  not  to  be  performed  at  all;  that  is,  unity  is  to 
be  left  unchanged;  thus  10°  or  10»-1  =  1,  2"  =  1,  log"  x  =  x, 
sin"  X  =  X,  etc. 

Exercises 

Select  as  unit  a  distance  1/2  inch  in  length  extending  to  the  right 
and  represent  graphically  each  of  the  following  expressions: 

1.  i  +  2i'  +  3i'  +  4i*  -f  . 

2.  t  +  i«  +  i*  +  i«  +  i'  +  . 

3.  i  +  i*  +  e  +  i^  +  i'  +  i'^  +  . 

4.  i(i  +  i<  +  i'  +  i*  +  i'  +  ii2  +  .        . ). 

5.  i  +  i«  -f- 1'  +  2i^  +  i*  +  t"  +  i'  +  3i»  +  .    .    . 


§^07]  ,"  COMPLEX  NUMBERS  367 

207.  Conjugate  Complex  Numbers.  Two  complex  numbers 
are  said  to  be  conjugate  if  they  differ  ohiy  in  the  sign  of  the  term 
containing  \/  —  1."  Such  are  x  +  iy  and  x  —  iy. 

Conjugate  imaginaries  have  a  real  sum  and  a  real  product. 

For  {x  +  yi)  +  {x  —  yi)  =  x  +  yi  +  x  —  yi, 

=.  X  +  X  +  yi  —  yi  =  2x. 

Likewise,  applying  the  ordinary  rules  of  algebra, 

{x  +  yi)  (x  —  yi)  =  x^  —  yH''  =  a;^  +  j/^ 

It  is  well  to  note  that  the  product  of  two  conjugate  complex 
numbers  is  always  positive  and  is  the  sum  of  two  squares. 

This  fact  is  very  important  and  will  be  used  frequently.     Thus 
(3  -  4i)(3  +  4.1)  =  3^'+  42  =  25;  (1  +  i){l  -  i)  =  2; 
(cos  d  +  i  sin  9)  (cos  d  —  i  sin  6)  =  cos"  6  +  sin"  0  =  1;  etc. 

208.  The  sum,  product,  or  quotient  of  two  complex  numbers  is, 
in  general,  a  complex  number  of  the  typical  form  a  +  bi. 

Let  the  two  complex  numbers  be  a;  +  yi  and  u  +  vi. 

(1)  Their  sum  is  (x  +  yi)  +  (u  +  vi) 

=  (x  +  u)  +  {y  +  v)i 

by  the  laws  of  algebra.     This  last  expression  is  in  the  form  a  +  bi. 

(2)  Their  product  is  {x  +  yi)  (u  +  vi) 

=  x{u  +  vi)+  yi{u  +  vi) 
=  xu  +  xvi  +  yui  +  yvi' 
=  {xu  —  yv)  +  {xv  +  yu)i 

by  the  laws  of  algebra.     This  last  expression  is  in  the  form  a  +  bi 

(3)  Their  quotient  is 

X  +  yi  _  (x  +  yi){u  —  vi) 
u  +  vi        (m  +  vi)(u  —  vi) 

By  the  preceding,  the  numerator  is  of  the  form  a'  +  b'i.    By 
§207,  the  denominator  equals  m"  +  «".    Then  the  quotient  equals 
a'  +  b'i  a'  b'        . 


u^  +  v^        m"  +  w"   '    m"  +  »2 
by  distributive  law.    This  last  expression  is  of  the  form  a  +  bi. 


368        ELEMENTARY  MATHEMATICAL  ANALYSIS 


Exercises 

Reduce  the  following  expressions  to  the  typiqal  form  a  +  bi;  the 
student  must  change  every  imaginary  of  the  form  -y/  —  o  to  the  form 

1.  V  -  25  +  V~^^  +  V^^i2i  -  V^'ei  -  6i. 

2.  (2V~^^  +  3v'^)(4\/"-^3  -  5V^^). 

3.  (x  -  [2  +3i]){x  -  [2  -3i]). 

4.  (-5  +  12V^T)^.  6.  (vr+i)(-v/r^). 

5.  (3  -  4V^.)'.  7.  (Ve"-  V^~^)'. 

a  1 

8.        , .  12. 


2  1  -  i^ 

13.; 


"■   S+V  -2   •  (1   -  0'- 

10.  ,       'V  _  14.  l^.^^A 

11.  1  +V  15.  (2  +  sV^^n.^ 


i-i"  2  +  v^n"  ■ 

-„    o  +  a;i        a  —  xi 

lb.     ^ j ;• 

a  —  XI       a  +  x% 

209.  If  a  complex  number  is  equal  to  zero,  the  imaginary  and 
real  "parts  are  separately  equal  to  zero. 

Suppose  X  +  y  \/  —  1  =  0, 

X  and  y  being  real  numbers. 

Then  x  =    —  y  V  —  1. 

Now  it  is  absurd  or  impossible  that  a  real  number  should  equal 
an  imaginary,  except  they  each  be  zero,  since  the  real  and  imagi- 
nary scales  are  at  right  angles  to  each  other  and  intersect  only  at 
the  point  zero. 

Therefore  x  =  0  and  y  =  0. 

If  two  complex  numbers  are  equal,  then  the  real  parts  and  the 
imaginary  parts  must  be  respectively  equal. 

For  if  X  +  yi  =  u  +  vi 

then  (x  -«)  +  (?/  -  v)i  =  0. 


§2101 


COMPLEX  NUMBERS 


369 


Whence,  by  the  above  theorem, 


That  is, 


X  —  u  =  0  and  y  —  v=  0. 
a;  =  M  and  y  =  v. 


210.  Modulus.  Let  the  complex  number  x  +  yihe  constructed, 
as  in  Fig.  152,  in  which  OA  =  x  and  AP  =  yi.  Draw  the  line 
OP,  and  let  the  angle  AOP  be  called  0. 

The  numerical  length  of  OP  is  called  the  modulus  of  the  complex 
number  x  +  yi.  It  is  algebraically  represented  by  -y/x^  +  y^, 
or  by  the  symbol  \x  +  yi\.    Thus,  mod  (3  +  4*)  =  V9  +  16  =  5. 

The  student  can  easily  see  that  two  conjugate  complex  numbers 
have  the  same  modulus. 

If  2/  =  0,  the  mod  (x  +  yi)  =  \/^=  \x\,  where  the  vertical 
lines  indicate  that  merely  the  numerical,  or  absolute,  value  of 
X  is  called  for.  Thus  the 
modulus  of  any  real  number 
is  the  same  as  what  is  called 
the  numerical,  or  absolute 
value,  of  the  number.  Thus 
mod  (—  5)  =  5. 

211.  Amplitude.  In  Fig. 
152  the  angle  AOP  or  6  is 
called  the  argument,  or  ampli- 
tude, or  simply  the  angle,  of 
the  complex  number  x  +  yi. 
Putting  r  =  \^x^  +  y^  -  mod  {x  + 

y 


Fig. 


152. — Modulus  and  amplitude 
of  a  complex  number. 

yi)  =  ;x  +iy\,  we  have 


sin  6  = 


and 


cos  6 


X 

r 


Therefore, 


.•B  +  ?/i  =  r  cos  0  +  ir  sin  Q  =  r(cos  9  +  i  sin  9). 


(1) 


We  have  expressed  the  complex  number  x  +  yi  in  terms  of  its 
modulus  and  amplitude.  The  last  member  of  (1)  is  called  the 
polar  fonn  of  the  complex  number  {x  +  iy). 

To  put  3  —  4i  in  this  form,  we  have 

mod  (3  -  4i)  =  \/9  +  16  =  5;  sin  5  =  ^  =  -  |;  cos  S  =  -  =  f 
^  r  5  r       5 

24 


370  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§212 
Therefore, 

The  amplitude  d  is  tan-'  (  ~  o )  i  ^^^  is  in  the  fourth  quadrant. 

Why? 

It  is  well  to  plot  the  complex  number  in  order  to  be  sure  of  the 
amplitude  6.  It  avoids  confusion  to  use  positive  angles  in  all 
cases.  For  example,  to  change  3  —  \/3  i  to  the  polar  form,  plot 
the  point  (3,  —  \/3)  and  find  from  the  triangle  that  r  =  2  \/3  and 
9  =  330°.    Hence 

3  -  VS  i  =  2V3(cos  330°  +  i  sin  330°).         [< 

The  ampUtude  of  all  positive  numbers  is  0,  and  of  all  negative 
numbers  is  180°.  The  unit  expressed  in  terms  of  its  modulus  and 
amplitude  is  evidently  l(cos  0  +  i  sin  0). 

212.  Vector.  The  point  P,  Fig.  152,  located  by  OA  +  AP,  or 
X  +  yi,  may  also  be  considered  as  located  by  the  line  or  radius 
vector  OP;  that  is,  by  a  line  starting  at  0,  of  length  r  and  making 
an  angle  6  with  the  direction  OX.  A  directed  line,  as  we  are  now 
considering  OP,  is  called  a  vector.  When  thus  considered,  the  two 
parts  of  the  compound  operator 

r  (cos  5  +  i  sin  6)  (1) 

receive  the  following  interpretation :  The  operator  (cos  6  +  ism  6), 
which  depends  upon  B  alone,  turns  the  unit  Ijdng  along  OX 
through  an  angle  6,  and  may  therefore  be  looked  upon  as  a  versor 
of  rotative  power  6.  The  versor  (cos  6  -\-  i  sin  6)  is  often  abbre- 
viated by  the  convenient  symbol  cWd.  The  operator  r  is  a  tensor, 
which  stretches  the  turned  unit  in  the  ratio  1 :  r.  The  result  of 
these  two  operations  is  that  the  point  P  is  locaited  r  units  from  0 
in  a  direction  making  the  angle  6  with  OX. 

Thus,  the  operator  (cos  ^  +  i  sin  6)  is  simply  a  more  general 
operator  than  i,  but  of  the  same  kind.  The  operator  i  turns  a 
unit  through  a  right  angle  and  the  operator  (cos  0  -\-  i  sin  6)  turns 
a  unit  through  an  angle  B.  If  6  be  put  equal  to  90°,  cos  6-\-i  sin  6 
reduces  to  i. 


§213]  COMPLEX  NUMBERS  '  371 

For  d  =  0,       cos  6  +  i  sin  $  reduces  to       1 

6  =  90°,  cos  d  +  i  sin  d  reduces  to  i 
6  =  180°,  cos  6  -\-  i  sin  6  reduces  to  —  1 
6  =  270°,  cos  6  +  isin  6 reduces  to  —  i 

Since  3  —  4i  =  5(f  —  fi),  the  point  located  by  3  —  4i  may  be 
reached  by  turning  the  unit  vector  through  an  angle  8  = 
sin~'(—  4/5)  =  COS"' 3/5  and  stretching  the  result  in  the  ratio  1  :5. 

//  a  complex  number  vanishes,  its  modulus  vanishes;  and  con- 
versely, if  the  modulus  vanishes,  the  complex  number  vanishes. 

li  X  +  yi  =  0,  then  x  =  0  and  y  =  0,  hy  §210.  Therefore, 
Vx^  +  t/2  =  0.  Also,  if  Va;^  +  y^  =  0,  then  x^  +  y^  =  0,  and 
since  x  and  y  are  real,  neither  x'  nor  y^  is  negative,  and  so  their 
sum  is  not  zero  unless  each  be  zero. 

//  two  complex  numbers  are  equal,  their  moduli  are  equal,  but  if 
two  moduli  are  equal,  the  complex  numbers  are  not  necessarily  equal. 

li  X  -i-  yi  =  u  +  vi,  then  x  =  u and y  =  vhy  §210. 
Therefore,  V^^+^  =  Vu^  +  vK 

But  if         ■y/x'^  +  y'^  =  y/u^  +  v^,  obviously  x"^  need  not  equal 
u^  nor  y"^  =  v'. 

213.  Sum  of  Complex  Numbers.  Let  a  given  complex  number 
locate  the  point  A,  Fig.  153,  and  let  a  second  complex  number 
locate  the  point  B.  Then  if  the  first  of  the  complex  numbers  be 
represented  by  the  radius  vector  OA,  and  if  the  second  complex 
number  be  represented  by  the  radius  vector  OB,  the  sum  of  the 
two  complex  numbers  will  be  represented  by  the  diagonal  OC  of 
the  parallelogram  constructed  on  the  lines  OA  and  OB.  This  law 
of  addition  is  the  well-known  .law  of  addition  of  vectors  used  in 
physics  when  the  resultant  of  two  forces  or  the  resultant  of  two 
velocities,  two  accelerations,  or  two  directed  magnitudes  of  any 
kind,  is  to  be  found. 

The  proof  that  the  sum  of  the  two  complex  numbers  is  repre- 
sented by  the  diagonal  OC  is  very  simple.  Let  the  graph  of  the 
first  complex  number  be  ODi  +  DiA  and  let  that  of  the  second  be 
OD2  -f-  DiB.  To  add  these,  at  the  point  A  construct  AE  =  ODi 
and  EC  =  D^B.  Then  the  sum  of  the  two  complex  numbers  is 
geometrically  represented  by  OJ)^  +  BiA  +  AE  +  EC,  or  by  the 


372        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§214 

radius  vector  OC  which  joins  the  end  points.  Since,  by  construc- 
tion, the  triangle  AEC  is  equal  to  the  triangle  OD^B,  AC  must  be 
equal  and  parallel  to  OB,  and  the  figure  OACB  is  a  parallelogram. 
OC,  which  represents  the  required  sum,  is  the  diagonal  of  this 
parallelogram,  which  we  were  required  to  prove. 


Di  Di  Di 

Fig.  153. — Sum  of  two  complex  numbers. 


Exercises 

Mnd  algebraically  the  sum  of  the  following  complex  numbers,  and 
construct  the  same  by  means  of  the  law  of  addition  of  vectors. 

1.  (1  +  2i)  +  (3  +  4i).  4.  (3  -  4i)  -  (3  +  4i). 

2.  (1  +  i)  +  (2  +  i).  5.  (-2  +  i)  +  (0  -  ti). 

3.  (1  -  i)  +  (1  +  2i).  6.  (-  1  +  i)  +  (3  +  i)  +  (2  +  2i). 

7.  (2  -  i)  +  (-  2  +  i)  +  (1  +  i)- 

8.  Find  the  modulus  and  ampHtude  ^in  degrees  and  minutes)  of 
2(cos  30°  .+  i  sin  30°)  +  (cos  45°  +  i  sin  45°). 

9.  By  the  parallelogram  of  vectors,  show  that  the  sum  of  two  con- 
jugate complex  numbers  is  real. 

10.  If  ij  be  the  sum  of  the  complex  numbers  Zi'=  xi  -|-  iyi,  Z8  = 
Xi  -\-iyi,  «a  =  Sa  +  Vii,  etc.,  show  that  —R,  zi,  zj,  23, .  .  .  form  the 
sides  of  a  closed  polygon. 

214.  Product  of  Complex  Nximbers.  The  product,  of  two  or 
more  complex  ^umbers  is  a  complex  number  whose  modulus  is'>the 


§214] 


COMPLEX  NUMBERS 


373 


product  of  the  moduli  and  whose  amplitude  is  the  sum  of  the  ampli- 
tudes of  the  com,plex  numbers. 
Let  the  complex  numbers  be 

^1  =  xi  +  y-ii  =  ri  (cos  Q\  +  i  sin  6i) 

Z2  =  a;2  +  2/21  ==  rj  (cos  02  +  i  sin  ^2),  etc. 

By  actual  multiplication,  we  get 

«i22  =  rir-2  [(cos  01  cos  02  —  sin  0i  sin  02)  + 

(sin  01  cos  02  +  cos  0i  sin  di)i\  =  rir^  [cos  (0i  +  02)  +  i sin  (0i  +  di)] 
Whence  it  is  seen  that  rir2  is  the  modulus  of  the  product -and 
(01  +  02)  is  the  amplitude. 


(2  +  2»)(v7+i) 


Fig.  154. — Product  of  two  pomplex  numbers. 

The  above  theorem  is  illustrated  by  Fig.  154.  If  the  two  given 
complex  numbers  be  represented  by  their  vectors  OPi  and  OPt,  their 
product  will  be  represented  by  the  vector  OP3  whose  direction  angle 
is  the  sum  of  the  amplitudes  of  the  two  given  factors,  and  whose 
length  OP3  is  the  product  of  the  lengths  OPi  and  OP2. 

The  figure  represents  the  product  (2  +  2i)  {y/z  +  i).  Expressed  in 
terms  of  modulus  and  ampUtude  these  may  be  written, 

-s/3'+   i  =  2(cos  30°  +  i  sin  30°) 
2  +  21  =  2v^(cos  45°  +  i  sin  45°) 


374        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§215 

Hence,     ri  =  2,      rj  =  2\/2,      Bi  =  30°,      6,  =  45° 
Therefore       (2  +  2i)(V3  +  i)  =  4v'2  (cos  75°  +  i  sin  75°) 


Exercises 

Find  the  moduli  and  amplitudes  of  the  following  products,  and 
construct  the  factors  and  products  graphically.  Take  a  positi-i/e  angle 
for  the  amplitude  in  every  case. 

1.  (1  +  \/3t)(2-\/3  +  2i).  4.  (1  +  iy.  _ 

2.  (2  +  W3i){2  +  2i).  5.  (2  -  2v'3i)(\/3  +  Si). 

3.  (V3  +  3i)(2  -  2i).  6.  (1  -  iy. 

7.  (1  +  i)\l  -  i)K 

8.  2  (cos  15°  +  i  sin  15°)  X  3  (cos  25°  +  i  sin  25°). 

Find  numerical  result  by  use  of  slide  rule  or  trigonometric  tables. 

9.  2(cos  10°  +  i  sin  10°)  X  (1/3) (cos  12°  +  i  sin  12°)  X 
6(co3  8°  +isin8°). 

10.  Find  the  value  of  4-\/2  (cos  75°  +isin75°)  +  (Vs  +  i). 

216.  Quotient  of  Two  Complex  Numbers.  The  quotient  of 
two  complex  numbers  is  a  complex  number  whose  modulus  is  the 
quotient  of  the  moduli  and  whose  amplitude  is  the  difference  of  the 
amplitudes  of  the  two  complex  numbers.  Let  the  complex  numbers 
be 

Si  =  Xi  +  yii  =  ri(cos  Q\-\-  i  sin  9i) 

Zi  =  Xi  +  yii  =  rjCcos  di  +  i  sin  62). 
We  have 

zi  _  ri(cos  Bi  +  i  sin  gi)(cos  dj  —  i  sin  6i) 
02      raCcos  02  +  i  sin  S2)(cos  82  —  i  sin  0i) 

^  ri[oos  {di  -  62)  +i  sin  (9i  -  gg)] 
r2  (cos''  ^2  +  sin^  62) 

=  -[cos  (^1  -■^2)  +  i  sin  (^i  -  62)]- 
r2 

Whence  it  is  seen  that  —  is  the  modulus  of  the  quotient  and 

(.01  —  02)  is  the  ampUtude. 

In  Fig.  155,  the  complex  number  represented  by  the  vector  OPi 
when  divided  by  the  complex  number  represented  by  OP2  yields 
the  result  represented  by  OP3,  whose  length  ri/rais  found  by  dividing 
the  length  of  OPi  by  the  length  of  OP2,  and  whose,  direction  angle 


§216] 


COMPLEX  NUMBERS 


376 


is  the  difference  (ffi  —  61)  of  the  amplitudes  of  OPi  and  OPi. 
The  figure  is  drawn  to  scale  for  the  case: 


5  (cos  60°  +  i  sin  60°) 
2  (cos  20°  +  i  sin  20°) 


=  (2.5)  (cos  40°  +  i  sin  40°) 


Fig.  155. — Quotient  of  two  complex  numbers. 


Exercises 


Find  the  quotient  and  graph  the  results  in  each  of  the  following 
exercises.  Always  take  ampUtudes  as  positive  angles  and  if  9j  >  61, 
take  9i  +  360°  instead  of  9i. 

1.  (1  +  \/3i)  -^  (V2  +  V2i).   3.  (SVS  -3i)  -i-  (  -  1  +  \/3i). 

2.  (i  +  iVSi)  -^  (^2  -  V2i).    4.  (1  -  VSi)  -h  i. 
6.  2(oos  36°  +  i  sin  36°)  -r-  5(cos  4°  +  i  sin  4°;. 

6.  1.2(cos  48°  +  i  sin  48°)  h-  [2(cos  15°  +  i  sin  15°) 

3(cos  9°  +  i  sin  9°;]. 
„    [4  +  (4/3)-v/3i]  (2  +  2Vdi) 

8  +  8i 
8.  Express  in  terms  of  a,  b,  e,  d,  the  ampUtude  of  (a  +  bi)  + 
(c  +  di).  I 

216.  De  Moivre's  Theorem.  As  a  special  case  of  §214  consider 
the  expression 

(cos  d  +  i  sin  S)» 

where  n  is  a  positive  number. 


376        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§216 

This  being  the  product  of  n  factors  like  (cos  e  +  i  sin  e),  we 
write,  by  means  of  §214, 

(cos  S  +  i  sin  6)  (cos  0  +  i  sin  S) 

=  [cos(0+e+  .   )+isin((9+e+   ...)], 

or 

(cos  5  +  i  sin  6)"  =  (cos  nd  +  i  sin  nd),  (1) 

which  relation  is  known  as  De  Moivre's  theorem. 

De  Moivre's  theorem  holds  for  fractional  values  of  n.    For,  first 
consider  the  expression 

(cos  e  +  i  sin  e)^^\ 

where  the  power  1/t  of  aos  d  -\- i  sin  B  is,  by  definition,  an 
operator  such  that  the  <th  power  of  the  expression  equals 
cos  0  +  i  sin  B. 

a 

Put  B  =  t(i>,  SO  that  4>  —  ~t 

Then    (cos  B  +  i  sin  9)'/'  =  (cos  tij)  +  i  sin  tij))^'^ 

=  [(cos  ^  +  i  sin  </.)']i'"  by  (1) 

=  cos  <j>  +  i  sin  <p 

=  cos  1  +  I  sin  1  •  (2) 

Next  consider  the  case  in  which  n  =  j.    We  know 

(cos  B  +  i  sin  BY''  =  [(cos  fl  +  i  sin  B)')]^'' 

=  (cos  sB+  i  sin  sfl)!/'    by  (1) 

=  cosy +i  sin  y    by  (2).  (3) 

Likewise  the  theorem  may  be  proved  for  negative  values  of  n. 
Illustkation  1.     Find  (3  +  i  \/3)*. 

Write  3  +  i  VS  =  2  V3(cos  30"  +  i  sin  30°). 

Then,  by  De  Moivre's  theorem, 

(3  +  i  VS)*  =  144(cos  120°  +  i  sin_120°) 
=  144(  -  l/2+_^V3i) 
=  -72  +  72-v/3i 


§217]  COMPLEX  NUMBERS  377 

Illustration  2.    Knd  (2  +  2i)". 
Write  2  +  2i  in  the  form 

2  +  2i  =  2  \/2_(i  V2  +  i  V2i) 
(2  +  2i)'i  =  (2  V2)"(cos  45°  +  i  sin  45°)" 
=  (2  \/2)"(cos  495°  +  i  sin  495°) 
=  (2  •v/2)"(ooa  135°  +  i  sin  135°) 
=  (2V'2)"(  -  ^  ^2  +i\/2i) 
=  2i«  (  -  1  +  i). 

Exercises 

Evaluate  the  following  by  De  Moivre's  theorem,  using  trigonomet- 
ric table  or  slide  rule  when  necessary. 

1.  (8  +8\/3t)".V 6.  [1/2  +  (l/2)V3i]*. 

2.  •>y27  (cos  76°  -  i  sin  75°).        7.  (1  +  i)'. 

3.  -^125i.  8.  (-  2  +  2i)^. 

4.  [cos  9° +  i  sin  9°]".  9.  [(1/2)V3  -  (l/2)i]'. 

5.  (S+VSi)'-  

-10.  Find  value  of  (-1  +  V  -  3)=  +  (-1   -  V  -  3)'   by  De 
Moivre's  theorem. 

11.  Find  the  value  of  x^  -  2z  +  2  for  x  =  1  +  i.  

12.  If  ii  =  -  1/2  +  (1/2)^^^  and  J2  =  -  1/2  -  (1/2)  V  -  3, 
showthatjV  =  l.jV  =  l,ji'  =J2,h^  =Ju3i^"  =jV  =  l,jV"'^'=ii. 

217.  The  Roots  of  Unity.    Unity  may  be  written 

1  =  cos  0   +  i  sin  0 

1  =  cos  2ir  +  i  sin  2ir 

1  =  cos  47r  +  i  sin  Air 

1  =  cos  6t  +  i  sin  6ir 


and  so  on.  By  De  Moivre's  theorem  the  cube  root  of  any  of  these 
is  taken  by  dividing  the  amplitudes  by  3.  Therefore,  from  the 
above  expressions  in  turn,  there  results 

Vi  =  cos  0  +  i  sin  0  =  1 

•^1  =  cos  (27r/3)  +  i  sin  (27r/3)  =  cos  120°  +  i  sin  120° 

=  -l/2  +  i(l/2)  V3 
Vl  =  cos  (4ir/3)  +  i  sin  (47r/3)  =  cos  240°  +  i  sin  240° 

_  =  -  1/2  -  i(l/2)  V3 
Vl  =  cos  67r/3  +  i  sin  Qir/S  =  same  as  first,  etc. 


378        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§217 

Therefore  there  are  three  cube  roots  of  unity.  Since  these  are  the 
roots  of  the  equation  a;'  —  1  =  0,  they  might  have  been  found  by 
factoring,  thus 

x>  -  1  =  (x  -  1)  ix^  +  x+  1) 

=  ix-l){x  +  1/2  +  i  Vsi)  (^  +  1/2-4  V3i) 
The  three  roots  of  unity  divide  the  angular  space  about  the  point 
0  into  three  equal  angles,  as  shown  in  Fig.  156.  In  the  same 
way,  it  can  be  shown  that  there  are  four  fourth  roots,  five  fifth 
roots,  etc.,  of  unity  and  that  the  vectors  representing  them  have 
modulus  1  and  amplitudes  that  divide  equally  the  space  about  0. 


B 

\ 

~^ 

1          S 

f               ^ 

o   /\ 

D 

-ii\        / 

y' 

V          1 

r 

j^              \ 

c 

^ 

-^-^ 

Fig.  156. — The  cube  roots  of  unity. 

IlliTJStbatign  1.     Find  Vvl+3i 
Write  \/3  +  3i  in  the  form 

VS  +  3i  =  2-\/3(cos  60°  +  i  sin  60°) 

Hence,  by  De*  Moivre's  theorem, 

WZ  +  3i)^  =  y/Vi  (cos  30°  +  i  sm  30°) 
=  -v/T2[(l/2)^V3  +  (l/2)i] 
=  (1/2)1^108+  (1/2)^^12  i 

A  second  root  can  be  found  by  writing 

VS  +  3i  =  2V'3  [cos  (60°  +  360°)  -  i  ein  (60°  +  360°)] 


|218]  COMPLEX  NUMBERS  379 

siace  adding  a  multiple  of  360°  to  the  amplitude  does  not  change 
the  value  of  the  sin  and  cosine.  In  applying  De  Moivre's  theorem, 
there  results 

(VS  +  3i)>^  =  ■>yi2  (cos  210°  +i  sin  210°)     ' 
=  V^12  [  -  (l/2)v'3  -  (l/2)i] 

Illttstration  2.     Find  the  cube  root  of  —  \/2  +  \/2  i. 
We  write: 

-  \/2  +  i  V2  =  2  (cos  135°  +  i  cos  135°) 

=  2[cos  (135°  +  Ji360°)  +  i  cos  (135°  +  7i360°)]. 

in  which  n  is  any  integer.     Hence 

{-\/2+i  V2)^  =  \/2  [cos  (45°  +  ?il20°)  +  i  sin  (45°  +  nl20°)]^ 
=  ^  (cos  45°  +  i  sin  45°)      for  n  =  0 
=  -^2  (cos  165°  +  i  sin  165°)  forra  =  1 
=  -^2  (cos  285°  +  i  sin  285°)  for  n  =  2. 

These  are  the  three  cube  roots  of  the  given  complex  number.  For 
«  =  3  the  first  root  is  obtained  a  second  time. 

Exercises 

Find  all  the  indicated  roots  of  the  following: 

1.  (8  +  SVsi)^. 6.  (2  +  2i)^. 

2.  i^27(cos  75°  +  i  sin  75°).  6.  32^. 

3.  -^iMi.  7.  V/5I2. 

4.  (  -  2  +  2i)^. 

8.  Find  to  four  places  one  of  the  imaginary  7th  roots  of  +  1. 
Note:  Cos  51°  25.7'  +  i  sin  51°  25.7'  =  0.6235  +  0.7818i. 

218.*  Irrational  Numbers.  A  rational  ntunber  is  a  number  that 
can  be  expressed  as  the  quotient  of  two  integers.  All  other  real 
numbers  are  irrational.  Thus  V^j  \/5j  V^?,  ir,  e,  are  irrational 
numbers.  An  irrational  number  is  always  intermediate  in  value 
to  two  rational  numbers  which  differ  from  each  other  by  a  number 
as  small  as  we  please.    Thus 

1.414,      <  •v/2  <  1.415 
1.4142    <  \/2  <  1.4143 
1.41421  <  V2  <  1.41422,  etc. 


380        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§218 

It  is  easy  to  prove  that  \/2  cannot  be  expressed  as  the  quotient 
of  two  integers    For,  if  possible,  let 

V2=l,  (1) 

where  a  and  6  are  integers  and  r  is  in  its  lowest  terms.  Squaring 
the  members  of  (1)  we  have 

2  =^.  (2) 

This  cannot  be  true,  since  2  is  an  integer  and  a  and  6  are  prime 
to  each  other. 

An  irrational  number,  when  expressed  in  the  decimal  scale,  is 
never  a  repeating  decimal.  For,  if  the  irratiqn,al  number  could  be 
expressed  in  that  manner,  the  repeating  decimal  could  be  evalu- 
ated by  §120  in  the  fractional  form  ^  _    '  which,  by  definition 

of  an  irrational  number,  is  impossible.  On  the  contrary,  every 
rational  number  when  expressed  in  the  decimal  scale  is  a  repeating 
decimal.    Thus  1/3  =  0.33  .       .   and  1/4  =  0.25000.  .    . 

The  proof  that  ir  and  e  are  irrational  numbers  is  not  given  in 
this  book.^ 

The  student  should  not  get  the  idea  that  because  irrational  num- 
bers are  usually  approximated  by  decimal  fractions,  that  the 
irrational  number  itself  is  not  exact.  This  can  be  illustrated  by 
the  graphical  construction  of  ■\/2.  Locate  the  point  P  whose 
coordinates  are  (1,  1).  Call  the  abscissa  OD  and  the  ordinate  DP. 
Then  OP  =  y/%  OZ)  =  1,  and  DP  =  1.  It  is  obvious  that  the 
hypotenuse  OP  must  be  considered  just  as  exact  or  definite  as  the 
legs  OB  and  DP  The  notion  that  irrational  numbers  are  inexact 
must  be  avoided. 

The  process  of  counting  objects  can  be  carried  out  by  use  of  the 
primitive  scale  of  numbers  0,  1,  2,  3,  4,  .  .  .  The  other  numbers 
made  use  of  in  mathematics,  namely, 

(1)  positive  and  negative  numbers 

(2)  integral  and  fractional  numbers 

(3)  rational  and  irrational  numbers 

(4)  real  and  imaginary  numbers, 

^  See  Monographs  on  Modern  Mathematics,  edited  by  J.  W.  A.  Young. 


§219]  COMPLEX  NUMBERS  381 

may  be  looked  upon  as  classes  of  numbers  that  permit  the  opera- 
tions siibtraction,  division  and  evolution,  to  be  carried  out  under  all 
circumstances.  Thus,  in  the  history  of  algebra  it  was  found  that 
in  order  to  carry  out  subtraction  under  all  circumstances,  negative 
numbers  were  required;  to  carry  out  division  under  all  circum- 
stances, fractions,  were  required;  to  carry  out  evolution  of  arith- 
metical numbers  under  all  circumstances,  irrational  numbers 
were  required;  finally  to  carry  out  evolution  of  algebraic  numbers 
under  all  circumstances,  imaginaries  were  required.  It  will  be 
found  that  it  wiE  not  be  necessary  to  introduce  any  additional 
form  of  number  into  algebra;  that  is,  the  most  general  number 
required  is  a  number  of  the  form  a  4-  6i,  where  a  and  6  are  positive 
or  negative,  integral  or  fractional,  rational  or  irrational.  This  is 
the  most  general  number  that  satisfies  the  following  conditions: 

(a)  The  possibility  of  performing  the  operations  of  algebra  and 
the  inverse  operations  under  all  circumstances. 

(6)  The  conservation  or  permanence  of  the  fundamental  laws  of 
algebra:  namely,  the  commutative,  associative,  distributive,  and 
index  laws. 

Further  extension  of  the  number  system  beyond  that  of  complex 
numbers  leads  to  operators  which  do  not  obey  the  commutative 
law  in  multiplication;  that  is,  in  which  the  value  of  a  product  is 
dependent  upon  the  order  of  the  factors,  and  in  which  a  product 
does  not  necessarily  vanish  when  one  factor  is  zero.  Numbers  of 
this  kind  the  student  may  later  study  in  the  introduction  to  the 
study  of  electromagnetic  theory  under  the  head  of  "Vector  ^ 
Analysis"  or  in  the  subject  of  "Quaternions.''  Such  numbers  or 
operators  do  not  belong  to  the  domain  of  numbers  we  are  now 
studying. 

219.*  Simple  Periodic  Variation  Represented  by  a  Complex  Num- 
ber. Fluctuating  magnitudes  exist  that  follow  the  law  of  S.H.M. 
although,  strictly  speaking,  such  magnitudes  can  be  said  to  be  "sim- 
ple harmonic  motions"  in  only  a  figurative  sense.  For  example  we 
may  think  of  the  fluctuations  of  the  voltage  or  amperage  in  an  alter- 
nating current  as  following  such  a  law.  Thus  if  E  represent  the 
electromotive  force  or  pressure  of  the  alternating  current,  then  the 
fluctuations  are  expressed  by 

E  =  Eo  sin  oit 


382        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§219 

or  by 

E  =  Eo  sin  2irft, 

where/is  the  frequency  of  the  fluctuation.  Instead  of  S.H.M.  such  a 
variable  is  naiore  accurately  called  a  sinusoidal  varying  magnitude, 
although  for  brevity  we  shall  often  call  it  S.H.M.  The  graph  in  rec- 
tangular coordinates  of  such  a  periodic  function  is  often  called  a 
"wave,"  although  this  term  should,  in  exact  language,  be  reserved  for 
a  moving  periodic  curve,  such  a.sy  =  a  sin  (hx  —  kt). 
If  the  polar  representation 

p  =  a  sin  (ot  —  <i)  (1) 

of  the  sinusoidal  varying  magnitude  be  used,  then  the  graph  of  (1) 
is  a  circle  of  diameter  a  inclined  the  angular  amount  uU  to  the  left  of 
the  axis  OY,  as  is  seen  at  once  by  calUng  cot  =  9  and  aU  =  a  in  the 
equation  of  the  circle  p  =  o  sin  (9  —  a).  The  circle  can  be  drawn 
when  the  length  and  direction  of  its  diameter  are  known;  that  is,  the 
circle  is  completely  specified  when  a  and  the  direction  of  a  (told  by 
a)  are  given.  Therefore  the  simple  harmonic  motion  is  completely 
symbolized  by  a  vector  OA  of  length  a  drawn  from  the  origin  in  the 
direction  given  by  the  angle  uti.    The  direction  angle  of  the  vector  OA  is 

a  +  2>  or  ah  +  g- 

The  circle  on  the  vector  OA  is  located  or  characterized  equally 
well  if  the  rectangular  coordinates  (c,  d)  of  the  end  of  the  diameter 
of  the  circle  be  given.  But  the  complex  number  c  +  diis  represented 
by  a  vector  which  coincides  with  the  diameter  o  of  this  circle.  Hence 
we  may  represent  the  circle  by  the  complex  niunber  c  +  di.     Its 

modulus  is  a  =  ■\/c'  +  d''  and  its  amplitude  is  a  +  s.     Therefore  if  in 

(1)  we  take  o  =  \/c*  +  d-,  at,  =  a  and  the  variable  angle  at  =  9,  we 
can  completely  describe  the  S.H.M.  by  the  complex  number  c  +  di. 
In  the  theory  of  alternating  currents  the  sinusoidal  varying  current  or 
voltage  can  conveniently  be  represented  by  a  complex  number,  and 
that  method  of  representing  such  magnitudes  is  in  common  use. 

One  of  the  advantages  of  representing  S.H.M.  by  a  vector  or  by  a 
complex  number  is  the  fact  that  two  or  more  such  motions  of  like 
periods  may  then  be  compounded  by  the  law  of  addition  of  vectors. 
This  method  of  find^g  the  resultant  of  two  sinusoidal  varying  mag- 
nitudes of  like  periods  possesses  remarkable  utihty  and  simplicity. 

To  summarize,  we  may  say: 

(o)  A  siniisoidal  varying  magnitude  is  represented  graphically  in 


§219] 


COMPLEX  NUMBERS 


383 


polar  coordinates  by  a  vector,  which  by  its  length  denotes  the  amplitude 
and  by  its  direction  angle  with  respect  to  OY  denotes  the  epoch  angle. 

(6)  Sinusoidal  varying  magnitudes  of  like  periods  may  be  compounded 
or  resolved  graphically  by  the  law  of  parallelogram  of  vectors. 

If  two  sinusoidal  varying  magnitudes  of  like  periods  are  in  quad- 
rature (that  is,  if  their  epoch  angles  differ  by  90°),  their  relation, 
neglecting  their  epochs,  can  be  completely  expressed  by  a  single  com- 
plex number.     Thus  let  two  S.H.M.  in  quadrature 


and 


E„  =  113  sin  a{t  -  h) 
Ec  =  40  cos  a{t  -  ti) 


(2) 
(3) 


Fig.  157. — Composition  of  two  S.H.M.  in  quadrature  by  law  of 
addition  of  vectors. 


be  represented  by  the  circles  and  by  the  vectors  marked  OEo  and 
0B„,  Fig.  157.     Call  the  resultant  of  these  Ei.     Then 

Ei  =  113  sin  fc)(i  -  «i)  +  40  cos  u(«  -  <i)  (4) 

=  -\/402  -t-  113^  sin  w{t  -  ti) 
=  120  sin  w{t  -  ti),  (5) 

where  wij  is  measured  as  shown  in  Fig.  157.     Instead  of  representing 
(2)  and  (3)  in  the  polar  diagram  by  0E„  and  OEe  and  their  resultant 


DX 


384       ELEMENTARY  MATHEMATICAL  ANALYSIS      [§220 

by  OEi,  we  may  represent  (2),  (3),  and  (4)  in  the  complex  number 
diagram,  Fig.  158,  by  £?„,  lEc,  and  Ec  +  iEc,  respectively.  Since  the 
modulus  and  amplitude  of  £„  +  iEc  are  y/Eo'  +  &*  and  a,  respec- 
tively, and  since  the  epoch  angle  of  the  resultant  in  Fig.  157  is  ut2  = 
toil  —  a,  we  can  state  the  resultant  as  follows: 

//  we  have  given  two  S.H.M.'s  in  quadrature  and  take  the  amplitude 
oj  the  one  possessing  the  greater  epoch  angle  as  c  and  the  amplitude 
of  the  other  S.H.M.  as  d,  and  construct  the  complex  number  c  +  di, 
then  this  complex  number  c  +  di  completely  characterizes  both  of  the 
S.H.M.'s  and  their  resultant.  For,  we  can  determine  the  modulus  p 
and  the  amplitude  aoi  c  +  di  and  then  if  wti  is  the  epoch  angle  of  the 
moticm  with  amplitude  c,  the  epoch  angle  of  the  resultant  is  ati  —  a. 

If  we  consider  the  two  harmonic 
P  motions 

iEc  p  —  0,1  sin  ui{t  —  ti) 

and 

p  =  02  sin  ci)(t  —  ti), 

,   '  and  if  Ji  be  greater  than  ti,  the  first 

Fig.  158.— Complex  number  S.H.M.  reaches  its  maximum  value 
representation  of  the  facts  ji^  ^^e  second  reaches  its  maxi- 
shown   by  polar   diagram,  Fig.      ^  mi.     ^    ^  o  ti  at   •    ^i. 

147  ^  B        >       B     mum.     The  first  S.H.M.  is  there- 

fore said  to  lag  the  amount  (<i  —  tij 
behind  the  second  S.H.M.  That  is,  a  S.H.M.  represented  by  a  circle 
located  anticlockwise  from  a  second  circle  represents  a  S.H.M.  that 
lags  behind  the  second. 

220.*  Illustration  from  Alternating  Currents.  The  steady  current 
C  flowing  in  a  simple  electric  circuit  is  determined  by  the  pressure  or, 
electromotive  force,  E  and  the  resistance  R  according  to  the  equation 
known  as  Ohm's  law, 

^     r' 

or 

E  =  CR. 

E  is  the  pressure  or  voltage  required  to  make  the  current  C  flow 
against  the  resistance  R.  If  the  current,  instead  of  being  steady, 
varies  or  fluctuates,  then  the  pressure  CR  required  to  make  the  current 
C  flow  over  the  true  resistance  is  called  the  ohmic  voltage,  or  ohmic 
pressure.  But  a  changing  or  fluctuating  current  in  an  inductive 
circuit  sets  up  a  changing  magnetic  field  around  the  circuit,  from  which 
there  results  a  counter  electromotive  force,  or  choking  effect,  due  to  the 
changing  of  the  current  strength.     This  electromotive  force  is  called 


§220]  COMPLEX  NUMBERS  385 

the  reactive  voltage  or  reactive  pressure.    The  choking  effect  that  it 
has  on  the  current  is  known  as  the  inductive  reactance.     In  case  of  a 
periodically  changing  current  it  acts  alternately  with  and  against  the 
current.     Opposite  to  the  reactive  voltage  there  is  a  component  of  the 
impressed  voltage  that  is  consumed  by  the  reactance.     See  Fig.  159. 
The  pressure  which  is  at  every  instant  applied  to  the  circuit 
from  without  is  called  the  impressed  electromotive  force,  or  voltage. 
Of  the  three  pressures — namely,  the  impressed  voltage,  the  ohmic 
voltage  (consumed  by  the  resistance)  and  the  reactive  voltage  con- 
sumed by  the  inductive  reactance — any  one  may  always  be  regarded 
as  the  resultajit  of  the  other  two.     Hence,  if  in  a  polar  diagram  the 
pressures  be  represented  in  magnitude  and  relative  phase  by  the  sides 
of  a  parallelogram,  the  impressed 
voltage  may   be  regarded  as  the 
diagonal    of    a    parallelogram    of 
which  the  other  two  pressures  are 
sides.     Since,    however,    the    re- 
actance or  the  counter  inductive 
pressure  depends  upon  the  rate  of 
change  of  the  current,  it  lags,  in 
the  case  of  a  sinusoidal  current,,     Fiq.    159. — Complex  number 
90°  behind   the   true,   or   ohmic,       diagram  of  equation  5,  §220. 
voltage,  which  last  is   always  in 

phase  with  the  current.  The  pressure  consumed  by  the  counter  in- 
ductive pressure  therefore  leads  the  current  by  90°.  Thus,  in  the 
language  of  complex  numbers 

Ei=E„  +  iE„  (1) 

in  which 

Ei  =  impressed  pressure 

Eo  =  ohmic  pressure,  or  pressure  consumed  by  the  resistance 
Ec  =  counter  inductive  pressure,  or  the  pressure  consumed 
by  reactance. 

It  is  found  that  the  counter  inductive  pressure  depends  upon  a  con- 
stant of  the  circuit  L  called  the  inductance  and  upon  the  angular  veloc- 
ity or  frequency  of  the  alternating  impressed  pressure,  so  that 

E^  =  2ir/LC  =  wLC 
Hence  (1)  may  be  written 

Ei  =  RC  +  i2irfLC  (2) 

=  flC  +  wLC  (3) 


386        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§220 
The  modulus  of  the  complex  number  on  the  right  of  this  equation  is 


Considering,  then,  merely  the  absolute  value  \Ec\  and  \C\  of  pressure 
and  current,  we  may  write 

Id  =     ,    '^'1     „  (4) 

From  the  analogy  of  this  to  Ohm's  law, 

^     r' 

the  denominator  -\/i22  _|.  ^■iii  ig  thought  of  as  limiting  or  restricting 
the  current  and  is  called  the  impedance  of  the  circuit. 

Let  there  be  a  condenser  in  the  circuit  of  an  alternator,  but  let  the 
circuit  be  free  from  inductance.  Then  besides  the  pressure  con- 
sumed by  the  resistance,  an  additional  pressure  is  required  at  any 
instant  to  hold  the  charge  on  the  condenser.  If  K  be  the  capacity 
of  the  condenser,  it  is  found  that  that  part  of  the  pressure  consumed 

C  C 

in  holding  the  charge  on  the  condenser  is  ?r7v-'  °^  ~xr'  ^nd  is  in  phase 

position  90°  behind  the  current  C.  The  choking  effect  of  this  on  the 
current  may  be  called  the  condensive  reactance.  When  a  condenser 
is  in  the  circuit  in  addition  to  inductance,  the  total  pressure  con- 
sumed by  the  reactance  has  the  form : 

^"■^^  "  2^' 
and  the  complex  number  that  symbolizes  the  vector  is 

iP 
Ei  =RC  +  i2^fCL  -  ~.  (5) 

(see  Fig.  159). 

Further  illustrations  of  the  applications  of  complex  numbers  to 
alternating  currents  is  out  of  place  in  this  book.  The  illustrations  are 
merely  for  the  purpose  of  emphasizing  the  usefulness  of  these  numbers 
in  applied  science.  An  interesting  application  of  the  use  of  complex 
numbers  to  the  problem  of  the  steam  turbine  will  be  found  in  Stein- 
metz's  "Engineering  Mathematics,"  Page  33. 

Exercises 

1.  Draw  the  polar  diagram  and  complex  number  representation  of 
&  if  iJ  =  5,  C  =  21,/  =  60,  L  =  0.009,  K  =  0.005. 

2.  Draw  a  similar  diagram  if  Ei  =  100,  Eo  =  .90,  /  =  40,  L  =  0 .  008, 
K  =  0.003. 


CHAPTER  XIII 
LOCI 

221.  Parametric  Equations.  The  equation  of  a  plane  curve  is 
ordinarily  given  by  an  equation  in  two  variables,  as  has  been 
amply  illustrated  by  numerous  examples  in  the  preceding  chapters. 
It  is  obvious  that  a  curve  might  also  be  given  by  two  equations 
containing  three  variables,  for  if  the  third  variable  be  eliminated 
from  the  two  equations,  a  single  equation  in  two  variables  results. 
When  it  is  desirable  to  describe  a  locus  by  means  of  two  equations  in 
three  variables  the  equations  are  known  as  parametric  equations, 
as  has  already  been  explained  in  §84.  Two  of  the  variables  usu- 
ally belong  to  one  of  the  common  coordinate  systems  and  the 
third  is  an  extra  variable  called  the  parameter.  In  applied  science 
the  variable  time  frequently  occurs  as  a  parameter. 

The  parametric  equations  of  the  circle  have  already  been  writ- 
ten.   They  are 

X  =  a  cos  d,  (1) 

y  =  a  sin  6, 

where  the  parameter  d  is  the  direction  angle  of  the  radius  vector 
to  the  point  (a;,  y).  Likewise  the  parametric  equations  of  the 
elUpse  have  been  written 

X  =  a  cos  d,  (2) 

y  r  h  sin  d, 

and  those  of  the  hyperbola  have  been  written 

a;  =  a  sec  d,  (3) 

y  =  h  tan  0. 

In  harmonic  motion,  the  ellipse  was  seen  to  be  the  resultant  of 
the  two  S.H.M.  in  quadrature 

X  =  a  cos  (lit,  (4) 

2/  =  6  sin  cat. 

Here  the  parameter  t  is  time. 

387 


388        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§222 

222.  Problems  in  Loci.  It  is  frequently  required  to  find  the 
equation  of  a  locus  when  a  description  of  the  process  of  its  genera- 
tion is  given  in  words,  or  when  a  mechanism  by  means  of  which  the 
curve  is  generated  is  fully  described.  There  is  only  one  way  to 
gain  facility  in  obtaining  the  equations  of  curves  thus  described, 
and  that  is  by  the  solution  of  numerous  problems.  Sometimes 
it  is  best  to  seek  the  parametric  equations  of  the  curve,  but 
sometimes  the  ordinary  polar  or  Cartesian  equation  can  be  ob- 
tained directly.    The  following  problems  are  illustrative: 


Fig.  160. — Generation  of  so-called  "elliptic  motion." 

(1)  A  straight  line  of  constant  length  a  +  b  moves  with  its  ends 
always  sliding  on  two  fixed  lines  at  right  angles  to  each  other. 
Find  the  equation  of  the  curve  described  by  any  point  of  the 
moving  line.     (See  §84,  exercise  23.) 

In  Fig.  160,  let  AB  be  the  line  of  fixed  length,  and  let  it  so  move 
that  A  remains  on  the  Z-axis  and  B  remains  on  the  F-axis.  Let 
any  point  of  this  line  be  P  whose  distance  from  A  is  6  and  whose 
distance  from  B  is  a.  If  the  angle  X'AB  be  called  6,  then  PD, 
the  ordinate  of  P,  is 

y  =  b  sin  9 
and  OD,  the  abscissa  of  P,  is 

X  =  a  cos  6, 
Therefore  P  describes  an  ellipse  of  semi-axes  a  and  6. 


§222]  LOCI  389 

(2)  A  circle  rolls  without  slipping  within  a  circle  of  twice  the 
diameter.  Show  that  any  point  attached  to  the  moving  circle' 
describes  an  ellipse. 

Draw  the  smaller  rolling  circle  in  any  position  within  the  larger 
circle,  and  call  the  point  of  tangency  T,  as  in  Fig.  160.  Since 
the  smaller  circle  is  half  the  size  of  the  larger  circle,  the  smaller 
circle  always  passes  through  0,  and  the  line  Adjoining  the  points 
of  intersection  of  the  small  circle  with  the  coordinate  axes  is,  for 
all  positions,  a  diameter,  since  the  angle  AOB  is  a  right  angle. 

If  we  can  prove  that  the  arc  AT  =  the  arc  HT  for  all  positions 
of  T,  then  we  shall  have  shown  that  as  the  small  circle  rolls  from  an 
initial  position  with  point  of  contact  at  H,  the  end  A  of  the  diam- 
eter AB  slides  on  the  line  OX.  Since  B  lies  on  OY  and  since  AB 
is  of  fixed  length,  this  proves  by  problem  (1)  that  any  point  of  the 
small  circle  lying  on  the  particular  diameter  AB  describes  an 
ellipse. 

To  prove  that  arc  AT  =  arc  HT,  we  have  that  the  angle  HOT 

is  measured  in  radians  by     „„ —  •   The  angle  AO'T  is  measured  in 

arc  AT 
radians  by     q,^    ■    Since  Z  AO'T  =  2  Z  HOT,  we  have 

arc  AT  __      arc  HT 
O'A     "         OH    ' 

But,  OH  =  20' A.     Hence  a.TC  AT  =  arc  HT. 

We  can  now  prove  that  any  other  point  of  the  rolling  circle  de- 
scribes an  ellipse.  Let  any  other  point  be  Pi.  Through  Pi  draw 
the  diameter  JO'K.  The  above  reasoning  applies  directly,  re- 
placing A  by  J  and  H  by  N. 

It  is  easy  to  see  that  all  points  equidistant  from  the  center  of  the 
small  circle,  such  as  the  points  P,  and  Pi,  describe  ellipses  of  the 
same  semi-axes  a  and  6,  but  with  their  major  axes  variously  in- 
clined to  OH. 

(3)  Determine  the  curve  given  by  the  parametric  equations 

X  =  a  cos  2ut  (1) 

y  =  a  sin  ut.  (2) 

^"Circle"  is  here  used  in  the  sense  of  a  "disc"  or  circular  area  and  not  in  the 
Beose  of  a  "  oiroumf erence." 


390        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§222 

To  eliminate  t,  the  first  equation  may  be  written 

K  =  a  (1  -  2sin2a)0.  (3) 

From  the  second  equation,  sin  cot  =  ~.  Substituting  for  sin  ut  in 
(3), 

-  =  «(l-^>  (4) 

or 

2/^=-|^  +  f-  (5) 

This  curve  is  the  parabola  y^  =  mx,  the  special  location  of  ■which 
the  student  should  describe. 

(4)  Construct  a  graph  such  that  the  increase  in  y  varies  directly 
as  X. 

If  y  varied  directly  as  x,  then  y  would  equal  kx,  where  A;  is  any 
constant.  In  the  given  problem,  however,  the  increase  in  y  (and 
not  y  itself)  must  vary  in  this  manner.  Let  the  initial  value  of  y 
be  represented  by  z/o.  Then  the  gain  or  increase  of  y  is  repre- 
sented by  2/  —  2/0.    Hence,  by  the  problem, 

y  —  yo  =  kx.  (1) 

Since  t/o  is  a  constant,  (1)  is  the  equation  of  the  straight  line  of 
slope  k  and  F-intercept  j/o,  which  ordinarily  would  be  written  in 
the  form 

y  =  kx  +  2/0. 

(5)  Express  the  diagonal  of  a  cube  as  a  function  of  its  edge,  and 
graph  the  function. 

If  the  edge  of  the  cube  be  x,  its  diagonal  is  -v/a;"  +  x^  +  x^  or 
X  \/3.  If  the  diagonal  be  represented  by  y,  we  have  y  =\/3x, 
which  is  a  straight  line. 

(6)  A  rectangle  whose  length  is  twice  its  breadth  is  to  be  in- 
scribed in  a  circle  of  radius  a.  Express  the  area  of  this  rectangle 
in  terms  of  the  radius  of  the  circle. 

Let  the  rectangle  be  drawn  in  a  circle  whose  equation  is 
x^  +  y^  =  a'.  At  a  corner  of  the  rectangle  we  have  x  =  2y.  The 
area  A  of  the  rectangle  is  4xy,  or  8y^  since  x  =  2y.    From  the 


§222]  LOCI  391 

equation  of  the  circle  we  obtain  4y^  +  y^  =  a^  or  y^  =  a^/5. 
Hence 

A  =  (8/5)a2. 

If  A  and  a  be  graphed  as  Cartesian  variables,  the  graph  is  a 
parabola. 

(7)  A  rectangle  is  inscribed  in  a  circle.    Express  the  area  of  the 
rectangle  as  a  function  of  a  half  of  one  side. 

Here,  as  above, 

A  =  4x2/  =  4x  Va^  —  x^- 

The  student  should  graph  this  curve,  for  which  purpose  a  may  be 
put  equal  to  unity.  First  draw  the  semicircle  y  =  \/'o^  —  x'- 
For  X  =  1/6,  take  one-fifth  of  the  ordinate  of  this  semicircle.  For 
X  =  2/5,  take  two-fifths  of  the  ordinate  of  the  semicircle,  and  so  on. 
The  curve  through  these  points  is  y  =  x  s/ a^  —  x^,  from  which 
y  =  4x  \/a^  —  x''  can  be  had  by  proper  change  in  the  vertical  unit 
of  measure. 

Exercises 

1.  In  polar  coordinates,  draw  the  curves: 

p  =  2  cos  8  p  =  2  cos  9  +  1 

P  =  2  cos  9  —  1  p  =  2  cos  9  -1-  3. 

2.  On  polar  coordinate  paper  select  the  point  (1,  1).  (This  means 
the  point  whose  coordinates  are  one  centimeter,  and  one  radian.) 
Starting  at  this  point,  a  point  moves  so  that  the  radius  vector  of  the 
moving  point  is  always  equal  to  the  vectorial  angle.  Sketch  the 
curve.     Write  the  polar  equation  of  the  curve. 

3.  A  point  moves  so  that  one  of  its  polar  coordinates,  the  radius 
vector,  varies  directly  as  the  other  polar  coordinate,  the  vectorial 
angle.  Write  the  polar  equation  of  such  a  curve.  Does  the  curve 
go  through  the  point  (!',  1)? 

4.  A  polar  curve  is  generated  by  a  point  Which  starts  at  the  point 
(1,  2)  and  moves  so  that  the  increase  in  the  radius  vector  always 
equals  the  increase  in  the  vectorial  angle.  Write  the  equation  of  the 
curve. 

6.  A  polar  curve  is  generated  by  a  point  which  starts  at  the  point 
(1,  2)  and  moves  so  that  the  increase  in  the  radius  vector  varies  directly 
as  the  increase  in  the  vectorial  angle.     Write  the  equation  of  the  curve. 

6.  A  ball  is  thrown  from  a  tower  with  a  horizontal  velocity  of  10 


392        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§222 

feet  per  second.  It  falls  at  the  same  time  through  a  variable  distance 
given  by  s  =  16. 1<^,  where  t  is  the  elapsed  time  in  seconds  and  a  is 
in  feet.     Find  the  equation  of  the  curve  traced  by  the  ball. 

7.  The  point  P  divides  the  line  AB,  of  fixed  length,  externally  in 
the  ratio  a  :  6,  that  is,  so  placed  that  PA/PB  =  a/b.  If  the  line  AB 
move  with  its  end  points  always  remaining  on  two  fixed  lines  OX  and 
OK  at  right  angles  to  each  other,  then  P  describes  an  ellipse  of  semi- 
axes  a  and  b. 

8.  If  in  the  last  problem  the  lines  OX  and  OY  are  not  at  right 
angles  to  each  other,  the  point  P  still  describes  an  ellipse. 

9.  A  point  moves  so  as  to  keep  the  ratio  of  its  distances  from  two 
fixed  lines  AC  and  BD  constant.  Prove  that  the  locus  consists  of 
four  straight  hnes. 

10.  A  sinusoidal  wave  of  amplitude  6  cm.  has  a  node  at  +  5  cm. 
and  an  adjacent  crest  at  +  8  cm.     Write  the  equation  of  the  curve. 

11.  The  velocity  of  a  simple  wave  is  10  meters  per  second.  The 
period  is  two  seconds.     Find  the  wave  length  and  the  frequency. 

12.  A  polar  curve  passes  through  the  point  (1,  1)  and  the  radius 
vector  varies  inversely  as  the  vectorial  angle.  Plot  the  curve  and 
write  its  equation.  Consider  especially  the  points  where  the  vectorial 
angle  becomes  infinite  and  where  it  is  zero.  Sketch  the  same  func- 
tion in  rectangular  coordinates. 

13.  Rectangles  are  inscribed  in  a  circle  of  radius  r.  Express  by 
means  of  an  equation  and  plot:  (o)  the  area,  and  (6)  the  perimeter  of 
the  rectangles  as  a  function  of  the  breadth. 

14.  Right  triangles  are  constructed  on  a  line  of  given  length  h  as 
hypotenuse.  Express  and  plot:  (a)  the  area,  and  (6)  the  perimeter  as 
a  function  of  the  length  of  one  leg. 

16.  A  conical  tent  is  to  be  constructed  of  given  volume,  V.  Express 
and  graph  the  amount  of  canvas  required  as  a  function  of  the  radius 
of  the  base. 

16.  A  closed  cylindrical  tin  can  is  to  be  constructed  of  given  volume, 
V.  Plot  the  amount  of  tin  required  as  a  function  of  the  radius  of  the 
can. 

17.  A  rectangular  water-tank  lined  with  lead  is  to  be  constructed 
to  hold  108  cubic  feet.  It  has  a  square  base  and  open  top.  Plot 
the  amount  of  lead  required  as  a  function  of  the  side  of  the  base. 

18.  An  open  cylindrical  water-tank  is  to  be  made  of  given  volume, 
V.  The  cost  of  the  sides  per  square  foot  is  two-thirds  the  cost  of  the 
bottom  per  square  foot.     Plot  the  cost  as  a  function  of  the  diameter. 

19.  An  open  box  is  to  be  made  from  a  sheet  of  pasteboard  12  inches 
square,  by  cutting  equal  squares  from  the  four  comers  and  bending  up 


§223] 


LOCI 


393 


the  sides.    Plot  the  volume  as  a  function  of  the  side  of  one  of  the 
squares  out  out. 

20.  The  illumination  of  a  plane  surface  by  a  luminous  point  varies 
directly  as  the  cosine  of  the  angle  of  incidence,  and  inversely  as  the 
square  of  the  distance  from  the  surface.  Plot  the  illumination  of  a 
point  on  the  floor  10  feet  from  the  wall,  as  a  function  of  the  height 
of  a  gas  burner  on  the  wall. 

21.  Using  the  vertical  distances  between  corresponding  points  on 
the  curves  y  =  sin  t  and  y  =  —  sin  t  as  ordinates  and  the  vertical 
distances  between  corresponding  points  oi  y  =  2t  and  j/  =  t^  as  abscis- 
sas, find  the  equation  of  the  resulting  curve. 

223.  Loci  Defined  by  Focal  Radii.  A  number  of  important 
curves  are  defined  by  imposing  conditions  upon  the  distances  of 
any  point  of  the  locus  from  two  fixed  points,  called  foci. 


Pig.  161. — The  lepmiscate. 

(1)  A  point  moves  so  that  the  product  of  its  distances  from  two 
fixed  points  is  constant.  Find  the  equation  of  the  path  of  the  par- 
ticle. Let  the  two  fixed  points  Fi  and  Fi,  Kg.  161,  be  taken  on  the 
X-axis  the  distance  o  each  side  of  the  origin.  Call  the  distances  of  P 
from  the  fixed  points  ri  and  rj.  Then  the  variables  ri  and  rj  in  terms 
of  ::;  and  y  are 

ri'^  =  y'  +  (x  -  a)2 


Hence 


ri'  =y'  +  (x  +  a)2. 
nW  =  [y^  +  (a;  -  o)"]  [y^  +  {x  +  a)']. 


(1) 
(2) 


Calling  the  constant  value  of  tiTi  =  c',  we  have,  as  the  Cartesian 
equation  of  the  locus, 


y'  +  (x-  ay]  [y^  +  (x  +  a)»]  =  c\ 


(3) 


394        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§223 


Fig. 


162. — The  lemniscate  and  the 
Cassinian  ovals. 


which  may  be  written 

(2/2  +  a;2  +  a2)2  -  4a^'x^  =  c*  (4) 

(x2  +  y'y  +  2aV  +  20^2/2  +  a*  -  iaV  =  c*  (5) 

V        (a;2  +  2/2)2  ^  2a%x'  -  y')  +  c*  -  a'.  (6) 

If  c  =  a  the  curve  is  called 
the  lemniscate,  and  the  Car- 
tesian equation  reduces  to 

(x2+ 2/2)2  =  2a2(a;2  -  2/=^)-  (7) 

For  other  values  of  c  the 
curves  are  known  as  the 
Cassinian  ovals.  The  stu- 
dent will  show  that  when 
c  <  u,  the  curve  consists  of 
two  separate  ovals  surround- 
ing the  foci,  and  for  c  >  a 
there  is  but  a  single  oval. 
The  curves  are  shown  in  Fig.  162.  These  curves  give  the  form  of 
the  equipotential  surfaces  in  a  field  around  two  positively  or  two 
negatively  charged  parallel  wires. 

(2)  Construct  the  curve  such  that  the  ratio  of  the  distances  of  any 
of  the  curve  from  two  fixed  points  is  constant.  Let  the  two  fixed 
points  be  A  and  B,  Fig. 
163;  let  the  constant  ratio 
of  the  distances  of  any  point 
of  the  curve  from  the  two 
fixed  points  be  n/J'2  =  mm. 
To  find  one  point  of  the 
locus,  draw  circles  from  A 
and  B  as  centers  whose  radii 
are  in  the  ratio  m/n.  Let 
these  circles  intersect  at  the 
point  P.  At  P  bisect  the 
angle  between  PA  and  PB 
internally  and  externally  by 
the  lines  PM  and  PN  respectively.  The  line  AB  ia  then  divided 
at  M  internally  in  the  ratio  MA/ MB  =  m/n  and  externally  at  N 
in  the  ratio  NA/NB  =  m/n,  because  the  bisectors  of  any  angle  of 
a  triangle  divide  the  base  into  segments  proportional  to  the  adja- 
cent sides.     Since  the  external  and  internal  bisectors  of  any  angle 


Fig.    163. — Construction  of 'the  curve 
ri/Ti  =  m/n,  or  the  circle  MPN. 


§224]  LOCI  395 

must  be  at  right  angles  to  each  other,  PM  is  perpendicular  to  PiV 
for  any  position  of  P.  Hence  the  locus  of  P  is  a  circle,  since  it  is  the 
vertex  of  a  right  triangle  described  on  the  fixed  hypotenuse  MN. 

If  a  large  number  of  circles  be  drawn  for  different  values  of  m/n,  and 
if  similar  circles  be  described  about  B,  then  these  circles  are  known 
as  the  dipolar  circles.  See  Fig.  164.  In  physics  it  is  found  that  these 
circles  are  the  equipotential  hnes  about  two  parallel  wires  perpendic- 
ular to  the  plane  of  the  paper  at  A  and  B  and  carrying  electricity  of 
opposite  sign. 


Fig.  164. — The  dipolar  circles,  or  a  family  of  circles  made  by 
drawing  ri/ra  =  e  for  various  values  of  e. 


Exercises 

1.  Draw  the  locus  satisfying  the  condition  that  the  ratio  of  the 
distances  of  any  point  from  two  fixed  points  ten  units  apart  is  2/3. 

2.  Draw  the  two  circles  which  divide  a  line  of  length  14  internally 
and  externally  in  the  ratio  3/4. 

224.  The  Cycloid.  The  cycloid  is  the  curve  traced  by  a  point 
on  the  circumference  of  a  circle,  called  the  generating  circle, 
which  rolls  without  slipping  on  a  fixed  line  called  the  base.  To 
find  the  equation  of  the  cycloid,  let  OA,'Yig.  165,  be  the  base,  P 
the  tracing  point  of  the  generating  circle  in  any  one  position,  and 
Q  the  angle  between  the  radii  SP  and  SH.  Since  P  was  at  0  when 
the  circle  began  to  roll, 

OH  =  ad, 


396        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§225 

if  a  be  the  radius  of  the  generating  circle.    Since  »=  OD  and 
y  =  DP,  we  have 


x=  OH-  SP  sin  e=  a(e  -  sin  6) 
y=  HS-  SP  cos  e=  a(i-  cos  (9). 


(1) 
(2) 


These  ar^  the  parametric  equations  of  the  curve.    For  most 
purposes  these  are  more  useful  than  the  Cartesian   equation. 


o  D      H     c  A  - 

Fig.  165. — Definition  of  the  cycloid. 

It  is  readily  seen  from  the  definition  of  the  curve,  that  the  locus 
consists  of  an  unlimited  number  of  loops  above  the  X-axis,  with 
points  of  contact  with  the  X-axis  at  intervals  of  2xa  (the  circum- 
ference of  the  generating  circle)  and  with  maximum  points  at 
X  -  Ta,  3xo,  etc. 


"=*'2Pl6  12  3  4  6 C  A 

Fig.  166. — Construction  of  the  cycloid. 

225.*  Graphical  Construction  of  the  Cycloid.  To  construct  the 
cycloid,  Fig.  166,  draw  a  circle  of  radius  1.15  inches  and  divide  the 
circumference  into  thirty-six  equal  parts.  Draw  horizontal  lines 
through  each  point  of  division  exactly  as  in  the  construction  of 
the  sinusoid,  Fig.  59-  Lay  off  uniform  intervals  of  1/5  inch  each 
on  the  X-axis,  marked  1,  2,  3,  .  .  .  Then  from  the  point  of 
division  of  the  circle  pi  lay  off  the  distance  01  to  the  right. 


§226] 


LOCI 


397 


From  pi  lay  off  02  to  the  right,  from  pa  lay  off  03  to  the  right, 
etc.  The  points  thus  determined  lie  on  the  cycloid.  The  number 
of  divisions  of  the  circumference  is  of  course  immaterial  except 
that  an  even  number  of  division  is  .convenient.  Further  the 
divisions  laid  off  on  the  base  OA  must  be  the  same  length  as 
the  arcs  laid  off  on  the  circle. 

Note  that  by  the  process  of  construction  above,  the  vertical 
distances  from  OX  to  points  on  the  curve  are  proportional  to 
(1  —  cos  6)  and  that  the  horizontal  distances  from  OY  to  points 
on  the  curve  are  proportional  to  (d—  sin  6). 

The  analogy  of  the  cycloid  to  the  sine  curve  is  brought  out  by 
Fig.  167.  A  set  of  horizontal  lines  are  drawn  as  before  and  also  a 
sequence  of  semicircles  spaced  at  horizontal  intervals  equal  to 


Fig.  167.— Analogy  of  the  cycloid  to  the  sinusoid. 


the  intervals  of  arc  on  the  circle.  The  plane  is  thus  divided 
into  a  large  number  of  small  quadrilaterals  having  two  sides 
straight  and  two  sides  curved.  Starting  at  0  and  sketching  the 
diagonals  of  successive  cornering  quadrilaterals  the  cycloid  is 
traced.  If,  instead  of  the  sequence  of  circles,  uniformly  spaced 
vertical  straight  lines  had  been  used,  the  sinusoid  would  have  been 
drawn;  The  sinusoid  on  that  account  is  frequently  called  the 
"companion  to  the  cycloid." 

226.  Epicycloids  and  Hj^ocycloids.  The  curve  traced  by  a 
point  attached  to  the  circumference  of  a  circle  which  rolls  without 
slipping  on  the  circimiference  of  a  fixed  circle  is  called  an  epi- 
cycloid or  a  hypocycloid  according  as  the  rolling  circle  touches 
the  outside  or  inside  of  the  fixed  circle.  If  the  tracing  point 
is  not  on  the  circumference  of  the  rolling  circle  but  on  a  radius 
or  radius  produced,  the  curve  it  describes  is  called  a  trochoid  if 


398        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§226 

the  circle  rolls  upon  a  straight  line,  or  an  epitrochoid  or  a  hypo- 
trochoid  if  the  circle  rolls  upon  another  circle. 

Exercises 

1.  Construct  a  cycloid  by  dividing  a  generating  circle  of  radius 
1.15  inches  into  twenty-four  equal  arcs  and  dividing  the  base  into 
intervals  3/10  inch  each. 

2.  Compare  the  cycloid  of  length  2ir  and  height  1  with  a  semi- 
ellipse  of  length  2ir  and  height  1. 

3.  Write  the  parametric  equations  of  a  cycloid  for  origin  C,  Fig.  165. 

4.  Write  the  parametric  equations  of  a  cycloid  for  origin  B,  Fig.  165. 
6.  Show  that  the  top  of  a  rolling  wheel  travels  through  space  twice 

as  fast  as  the  hub  of  the  wheel. 

6.  By  experiment  or  otherwise  show  that  the  tangent  to  the  cycloid 
at  any  point  always  passes  through  the  highest  point  of  the  generating 
circle  in  the  instantaneous  position  of  the  circle  pertaining  to  that 
point. 


CHAPTER  XIV 


THE  CONIC  SECTIONS 


227.  The  Focal  Radii  of  the  EUipse.  Draw  any  ellipse  with 
major  and  minor  circles  of  radii  a  and  6  respectively,  as  in  Fig. 
168.  Draw  tangents,  II'  and  KK',  to  the  minor  circle  at  the 
extremities  of  the  minor  axes  and  comJ)lete  the  rectangle  II'KK'. 


Properties  of  the  elUpse. 


The  points  Fi  and  Fi,  in  which  IK  and  I'K'  cut  the  major  axis,  are 
called  the  foci  of  the  ellipse.  Prom  any  point  on  the  ellipse  draw 
the  focal  radii  PFj  =  rj.  and  PF2  =  r^,  as  shown  in  the  figure. 
Represent  the  distance  OFi  =  OF 2  by  c.  Then  it  follows  from 
the  triangle  OIFi  that 

a^  =  b«  +  c^.  (1) 

This  is  one  of  the  fundamental  relations  between  the  constants  of 
the  ellipse.  > 

399 


400      ELEMENTARY  MATHEMATICAL  ANALYSIS        [§227 

From  the  triangles  PFiD  and  PFiD  there  follows: 

n^  =  (c  -  xY  +  v'  (2) 

r/  =  (c  +  xy  +  if.  (3) 

But  the  equation  of  the  ellipse  is 

b     , 

2/  =  -  Vo'  -  x\ 

or 

y^  =  ^,(a^  -  a;^).  (4) 

Substituting  this  value  of  y^  in  (2) 

ri'  =  c''  -  2cx  +  x^  +  -.(a''  -  x^)  (5) 

=  c^  -2cx  +  x^  +  ¥  -~i  a;^ 
or  by  (1) 

r,"  =  a^  -  2cx  +  a;^  [l  -  -^J .  (6) 

Substituting 

1  _  ^  -  «'  -  fe'  _  ^, 


we  obtain 


/i2/»2 


ri2  =  o2  -  2ca;  +  ^  (7) 

=  [•-"]""  <») 

Therefore 

Likewise,  from  (3),  by  exactly  the  same  substitutions,  there 
follows 

r2  =  a  +  "''■  (10) 

a 

From  (9)  and  (10),  by  addition, 

ri  +  Tz  =  2a.  (11) 

Hence  in  any  ellipse  the  sum  of  the  focal  radii  is  constant  and  equal 
to  the  major  axis, 


§228]  THE  CONIC  SECTIONS  401 

The  converse  of  this  theorem,  namely,  if  the  sum  of  the  focal 
radii  of  any  locus  is  constant,  the  curve  is  an  ellipse,  can  readily 
be  proved.  It  is  merely  necessary  to  substitute  the  values  of  ri 
and  Ti  from  (2)  and  (3)  in  equation  (11),  and  simplify  the  resulting 
equation  in  x  and  y;  or  first  square  (11)  and  then  substitute  ri  and 
r-2  from  (2)  and  (3).  There  results  an  equation  of  the  second 
degree  lacking  the  term  xy  and  having  the  terms  containing  x''  and 
y^  both  present  and  with  coefficients  of  like  signs.  By  §86,  such  an 
equation  represents  an  ellipse. 

Hence  the  ellipse  might  have  been  defined  as  the  locus  of  a 
point,  the  sum  of  whose  distances  from  two  fixed  points  is  constant. 

An  ellipse  can  be  drawn  by  attaching  a  string  of  length  2a  by 
pins  at  the  points  F\  and  F2  and  tracing  the  curve  by  a  pencil  so 
guided  that  the  string  is  always  kept  taut.  Or  better,  take  a 
string  of  length  2a  +  2c  and  form  a  loop  enclosing  the  two  pins; 
the  entire  curve  can  then  be  drawn  with  one  sweep  of  the  pencil. 

The  focal  radii  may  also  be  evaluated  in  terms  of  the  para- 
metric or  eccentric  angle  0.  The  student  may  regard  the  follow- 
ing demonstration  of  the  truth  of  equation  (11)  as  simpler  than 
that  given  above. 

Since  a;  =  a  cos  6,  and  y  =  h  sm  d 

equation  (2)  gives 

ri"  =¥  sin^  e  -h  (c  -  a  cos  df  (12) 

=  h^aia^e  +  c^  -  2accos0  +  o^  cos^  d.       (13) 

To  put  the  right  side  in  the  form  of  a  perfect  square,  write 
W  =  a^  -  c^.    Then 

ri'  =  a^  sin''  0  —  c^  sin''  B  +  c^  —  2ac  cos  d  +  a'  cos^  6 

=  o^  -  2oc  cos  e  +c'^  cos^  e.  (14) 


Whence 
Likewise 
Whence 


ri  =  a  —  c  cos  9.  (15) 

rj  =  a  -|-  c  cos  6.  (16) 


ri  -H  r2  =  2a. 

228.  The  Eccentricity.    The  ratio  c/a  measures,  in  terms  of  o  as 
unit,  the  distance  Pf  either  fopus  from  the  center  of  the  ellipse. 


26 


402       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§229 

This  ratio  is  called  the  eccentricity  of  the  ellipse.  In  the  triangle 
IFiO,  the  ratio  c/a  is  the  cosine  of  the  angle  FiOI,  represented  in 
what  follows  by  /3.    Calling  the  eccentricity  e,  we  have 

e  =  c/a  =  cos  |8.  (1) 

The  ellipse  is  made  from  the  major  circle  by  contracting  its  ordi- 
nates  in  the  ratio  m  =  b/a,  or  by  orthographic  projection  of  the 
circle  through  the  angle  of  projection 

a  =  cos~i  b/a. 

Hence,  as  companion  to  (1)  we  may  write 

m  =  b/a  =  cos  a  =  sin  |8.  (2) 

229.  The  Ratio  Definition  of  the  Ellipse.  In  Fig.  168,  let  the 
tangents  to  the  major  circle  at  I  and  I'  be  drawn.  Draw  a 
perpendicular  to  the  major  axis  produced  at  the  points  cut  by 
these  tangents.  These  two  lines  ai'e  called  the  directrices  of  the 
ellipse. 

We  shall  prove  that  the  ratio  PFi/PH  (or  PFt/PH')  is  constant 
for  all  positions  of  P.    From  §227,  equation  (9)  or  (15), 

ri  =  a  —  c  cos  d,  (1) 

/?•  (2) 


(3) 


From  the  figure, 

ON  =  a  sec  ION  =  a 

But 

cos  |8  =  c/a. 

Hence, 

ON  =  a^/c. 

But 

PH  =  0N  -  X. 

Therefore 

PH  =  a'^/c  -  a  cos  6. 

Hence,  from  (1)  and  (4), 

r\ 

=  PI 

^  /PH  -     "'-''  cos  9 

PH 

a'^/c  —  acos  e 

c  a  —  c  cos  6 

(4) 


a   a—  c  cos  d 
or 

PFi/PH  ^  c/a  =  e  =  cos  |S.        .    ,    ,   .     (6) 


§229]  THE  CONIC  SECTIONS  403 

A  similar  proof  holds  for  the  other  focus  and  directrix.  Thus, 
for  any  point  on  the  ellipse  the  distance  to  a  focus  bears  a  fixed 
ratio  to  the  distance  to  the  corresponding  directrix.  From  (5), 
the  ratio  is  seen  to  be  less  than  unity. 

Assuming  the  converse  of  the  above,  the  ellipse  might  have  been 
defined  as  follows :  The  ellipse  is  the  locus  of  a  point  whose  distance 
from  a  fixed  point  (called  the  focus)  is  in  a  constant  ratio,  less  than 
unity,  to  its  distance  from  a  fixed  line  (called  the  directrix). 

If,  in  any  ellipse,  c  =  0,  it  follows  that  b  must  equal  a  and  the  el- 
lipse reduces  to  a  circle.  If  c  is  nearly  equal  to  a,  then  from  the 
equation 

a2  =  62  +  c^, 

it  foljows  that  the  semi-minor  axis  &  must  be  very  small.     That  is, 
for  an  eccentricity  nearly  unity  the  ellipse  is  very  slender. 

If  the  sun  be  regarded  as  fixed  in  space,  then  the  orbits  of  the 
planets  are  ellipses,  with  the  sun  at  one  fobus.  (This  is  "Kepler's 
First  Law.")  The  eccentricity  of  the  earth's  orbit  is  0.017.  The 
orbit  of  Mercury  has  an  eccentricity  of  about  0.2,  which  is  greater 
than  that  of  any  other  planet. 

Exercises 

Find  the  eccentricities  and  the  distance  from  center  to  foci  of  the 
following  elUpses: 


1.  a;V9  -I-  2/74  =  1.  i.  2y    =  Vl  -  x". 

2.  2/  =  (2/3.)-\/36  -  xK  5.  Qx"  +  IQy'  =  14. 

3.  25a;2  +  iy'  =  100.  6..2a;2  +  3^^  =  1. 

Find  the  equation  of  the  ellipse  from  the  following  data: 

7.  e  =  1/2,  a  =  4.     Draw  this  ellipse. 

8.  c  =  4,  a  =  .5. 

9.  ri  =  6  -  2a;/3,  ri  =  6  +  2a;/3. 

10.  ri  =  5  —  4  cos  0,  ri  =  5  -j-  4:  cos  6. 

Solve  the  following  exercises: 

11.  Find  the  eccentricity  of  the  ellipse  made  by  the  orthographic 
projection  of  the  circle  x'  +  y^  =  a'  through  the  angle  60°. 

,   12.  The  angle  of  projection  of  a  circle  x'  +  y'  =  a'  by  which  an 


404      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§230 

ellipse  is  formed  is  a.    Show  that  the  eccentricity  of  the  ellipse  is 
sia  a. 

13.  A  circular  cylinder  of  radius  5  is  cut  by  a  plane  making  an 
angle  30°  with  the  axis.     Find  the  eccentricity  of  the  elliptic  section. 

14.  If  the  greatest  distance  of  the  earth  from  the  sun  is  92,500,000 
miles,  find  its  least  distance.     (Eccentricity  of  earth's  orbit  =  0.017.) 

16.  In  the  ellipse  a;*/25  +  y'/16  =  1,  find  the  distance  between 
the  two  directrices. 

16.  Write  the  equation  of  the  ellipse  whose  foci  are  (2,  0),  (  —  2,  0), 
and  whose  directrices  are  x  =  5  and  a;  =  —  5. 

17.  Prove  equation  11  §227  by  transposing  one  radical  in: 

V(.x+c)^+y^  +  V{x  -  c)2  +  y'  =  2o 

squaring,  and  reducing  to  an  identity. 

18.  Obtain  the  equation  of  the  ellipse  from  the  definition  at  the  top 
of  page  403. 

230.  The  Latus  Rectum.  The  double  ordinate  through  the 
focus  is  called  the  latus  rectum  of  the  ellipse.  The  value  of  the 
semi-latus  rectum  is  readily  formed  from  the  equation 

y  =  (b/a)  Va"  -  a;' 

by  substituting  c  for  x.    If  I  represents  the  corresponding  value 

oiy,  

I  =  (b/a)  Va^  -  c2  =  6Va  (1) 


(2) 


since  a^  —  c'^ 

=  6^. 

Hence  the  entire 

lnius 

by 

21  =  ?^^ 

a 

Equation  (1) 

may 

be  also  be  written 

l  =  bVi- 

c-'/a- 

=  b  Vi- 

e\ 

(3) 

In  Fig.  168  the  distances  AF,  AN,  ON,  OB,  OF,  FN  may 
readily  be  expressed  in  terms  of  a  and  e  as  follows  in  equations  (4) 
to  (10).  The  addition  of  the  formulas  (11),  (12),  (13)  brings  into 
a  single  table  all  the  important  formulas  of  the  ellipse. 

AFi  =  a  -  c  =  a(i  -  e)  (4) 


§230] 


THE  CONIC  SECTIONS 

405 

e               e 

(5) 

ON  =  a  sec  ^  =  1 

(6) 

e  =  cos  |8 

(7) 

OB  =  b  =  a  sin  /3  =  a  Vi  -  e^ 

(8) 

OFi  =  c  =  ae 

(9) 

FiN  =  OiV  -  c  =  a(i  -  e=)/e 

(10) 

1  =  bVa  =  a(i  -  e'^) 

(11) 

Ti  =  a  —  ex  =  a  —  X  cos  |3 

(12) 

ra  =  a  +  ex  =  a  +  X  cos  j3 

(13) 

Exercises 

1.  Find  the  value  in  miles  of  OFi  for  the  case  of  the  earth's  orbit. 

2.  Find  the  equation  of  an  ellipse  whose  minor  axis  is  10  units  and 
in  which  the  distance  between  the  foci  is  10. 

3.  In  the  ellipse  y  =  (2/3)\/36  -  x'  find  the  length  of  the  latus 
rectum  and  the  value  of  e. 

4.  The  eccentricity  of  an  ellipse  is  3/5  and  the  latus  rectum  is  9 
units.     Find  the  equation  of  the  ellipse. 

6.  In  (a)  X'  +  4v2  =  4  and  (6)  2x'  -|-  32/^  =  6  find  the  latus  rfectum, 
the  eccentricity,  and  the  distances  ON  and  AF. 

6.  Determine  the  eccentricities  of  the  ellipses, 

(o)  j/2  =  4s  -  (l/2)x2  (b)  j/2  =  4x  -  2x\  . 

7.  Find  the  value  of  /3  for  the  earth's  orbit.  (Use  the  S  functions 
of  the  logarithmic  table.) 

8.  Find  the  equation  of  an  ellipse  whose  latus  rectum  is  2  units  and 
minor  axis  is  4. 

9.  The  distance  from  the  focus  to  the  directrix  is  16  units.  The 
distance  from  the  vertex  to  the  nearest  focus  is  6  units.  Find  the 
equation  of  the  ellipse. 

10.  The  axes  of  an  ellipse  are  known.     Show  how  to  locate  the  foci. 

11.  In  an  ellipse  a  =  25  feet,  e  =  0.96.    What  are  the  values  of 
c  and  6? 

12.  For  a  certain  comet  (Tempel's)  the  semi-major  axis  of  the' 
elliptic  orbit  is  3.5,  and  c  =  1.4  on  a  certain  scale.     For  another 


406       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§231 

comet  (Enke's)  o  =  2.2,  e  =  0.85.    Sketch  the  curves,  taking  3  cm. 
or  1  inch  as  unit  of  measure. 

13.  If  i  =  7.2  and  e  =  0.6,  find  c,  a,  6. 

14.  An  ellipse,  with  center  at  the  origin  and  major  axis  coinciding 
with  the  X-axis,  passes  through  the  points  (8,  3)  and  (6,  4).  Find  the 
axes  of  the  ellipse. 

231.  Focal  Radii  of  the  Hjrperbola.  Construct  a  hyperbola 
from  auxiliary  circles  of  radii  o  and  b,  then  the  transverse  axis  of 
the  hyperbola  is  2a  and  the  conjugate  axis  is  26.  Unlike  the  case 
of  the  ellipse,  b  may  be  either  greater  or  less  than  a.  As  previously 
explained,  the  asymptotes  are  the  extensions  of  the  diagonals  of  the 
rectangles  BTAO,  BT'A'O.     From  the  points  /,  /',  in  which  the 


\^ 

H 

nX 

t' 

Hy^ 

W^ 

f 

i 

r 

\m 

y/ 

Fo      . 

A' 

\ 

V 

>y 

A 

I 

^Fi 

D 

y^ 

(2) 

B'  ^y 

< 

Fig.  169.— Properties  of  the  hyperbola. 


asymptotes  cut  the  a-circle,  draw  tangents  to  the  o-circle.  The 
points  Fi,  F2  in  which  the  tangents  cut  the  axis  of  the  hyperbola 
are  called  the  foci.     See  Fig.  169. 

The  distance  OFi  or  OFa  is  represented  by  the  letter  c.    Then, 

since  the  triangles  FJO  and  OAT  are  equal,  FJ  must  equal  6,  so 

that  we  Jbave  the  fundamental  relation  between  the  constants  of 

"the  hyperbola 


a^  +  b^  =  c" 


(1) 


§232]  THE  CONIC  SECTIONS  407 

From  any  point  on  either  branch  of  the  hyperbola  draw  the  focal 
radii  PFi  and  PF2,  represented  by  ri  and  rj  respectively.  Then 
from  the  figure 

ri^  =  (x  -  c)'+y\  (2) 

But  from  the  equation  of  the  hyperbola 

y^  =  {V/a?){z^  -  a?),  (3) 

hence 

n^  =  (x  -  cY  +  Wix"  -  a?)  la"-  (4) 

=  {aH^  -  2a^cx  +  aH^  +  Vx^  -  a^V")  /a"  (5) 

=  {cH''  -  2a?ex  +  a")  /a?-  (6) 

=  {ex  -  o?)ya\  (7) 

Hence  r\  =  {c/a)x  —  a.  (8) 

In  like  manner  it  may  be  shown  that 

r2  =  {c/a)x  +  a.  (9) 

Hence  from  (8)  and  (9)  it  follows 

r2  —  Ti  =  2a.  (10) 

Thus^  in  any  hyperbola,  the  difference  between  the  distances  of  any 

point  on  it  from  the  foci  is  constant  and  equal  to  the  transverse  axis. 

The  above  relation  may  be  derived  in  terms  of  the  parametric 

angle  9.    Since  in  any  hyperbola  x  =  a  sec  8  and  y  =  b  tan  6, 

ri"  =  V  tan''  (9  +  (o  sec  0  -  cf 

=  W  tan^  6  +  a'  sec"  9  —  2ac  sec  0  +  c\ 

To  put  the  right-hand  side  in  the  form  of  a  perfect  square,  write 
¥  =  c^  -  a\     Then 


ri^  =  c'  sec'*  9  -  2ac  sec  9  +  a\ 

Therefore 

ri  =  0  sec  9  —  a. 

and 

I2  =  c  sec  9  +  a. 

(11) 

(12) 

232.  The  Ratio  Property  of  the   Hyperbola.     Through   the 
points  of  intersection  of  the  a-circle  with  the  asymptotes,  draw 


408      ELEMENTAKY  MATHEMATICAL  ANALYSIS       (§233 

IK,  and  I'K'.  These  lines  are  called  the  diiectrices  of  the 
hyperbola.  It  wiU  now  be  proved  that  the  ratio  of  the  distance  of 
any  point  of  the  hyperbola  frota  a  focus  to  its  distance  from  the 
corresponding  directrix  is  constant.    Adopt  the  notation 

c/a  =  sec  ;8  =  e.  (1) 

Then,  from  the  figure 

PFi/PH  =  ri/{x  -  ON)  =  ri/(a  sec  0  -  o  cos  ;8)  (2) 

Substituting  ri  from  (11)  above: 

PF^/PH  =      "^'l^-'*     ^  (3) 

'  a  sec  0  —  a  cos  j8  ^  ' 

=    c  sec  8  —  a  c 

(4) 


a  (^sec  e  -  -j 


which  proves  the  theorem.  The  constant  ratio  e  is  called  the 
eccentricity  of  the  hyperbola,  and,  as  shown  by  (1),  is  always 
greater  than  unity. 

Assuming  the  converse  of  the  above,  it  is  obvious  that  the  hyper- 
bola might  have  been  defined  as  follows:  The  hyperbola  is  the 
locus  of  a  point  whose  distance  from  a  fixed  point  (called  the  focus) 
is  in  a  constant  ratio,  greater  than  unity,  to  its  distance  from  a  fixed 
line  {called  the  directrix).    Proof  will  be  given  in  §234. 

233.  The  Latus  Rectum.  The  double  ordinate  through  the 
focus  is  called  the  latus  rectum  of  the  hyperbola.  The  value  of 
the  semi-latus  rectum  is  readily  found  from  the  equation 

y  =  (b/a)  Vx^  -  a^ 

by  snbstituting  c  f  or  a;.     HI  represent  the  corresponding  value  of  y, 

I  =  (b/a)  Vc'  -  a^  =  hya.  (1) 

Hence  the  entire  latus  rectum  is  represented  by 

2l  =  2bVa.  (2) 

Equation  (1)  may  also  be  written 

i  =  6  Ve'  -  1.  (3) 


§233] 


THE  CONIC  SECTIONS 


409 


In  Fig.  169  the  distances  AFi,  AN,  ON,  OB,  OFi,  FiN  may 
readily  be  expressed  in  terms  of  o  and  e,  as  follows  in  equations 
(4)  to  (8).  Collecting  in  a  single  table  the  other  important  for- 
mulas for  the  hyperbola,  we  have : 


AFi  =  c  —  a  =  a(e  —  i) 

(4) 

AN  =  AF^/e  =  a(e  -  i)/e 

(5) 

ON  =  a  cos  /3  =  a/e 

(6) 

e  =  sec  /3 

(7) 

OB  =  b=atan/3  =  a  Ve^  -  i 

(8) 

OFi  =   c  =  ae 

FiN 

=  c  -  OiV  =  ae  -  a/e  =  aCe^  -  i)/e 

(9) 

1  =  bVa  =  b  Ve^  -  1  =  a(e2  -  i) 

(10) 

ri  =  ex  —  a  =  X  sec  iS  —  a 

(11) 

Tz  =  ex  +  a  =  X  sec  |3  +  a 

(12) 

The  important  properties  of  the  hyperbola  are  quite  similar 
to  those  of  the  ellipse.  It  is  a  good  plan  to  compare  them  in 
parallel  columns. 


Ellipse 


Hyperbola 


1.  Definition  of  Foci  and  Focal 
Radii 

2.  a2  =  b2  +  c2 

3.  ri  +  ra  =  2a 

4.  Eccentricity,  e  =  -  =  cos  /3 

5.  Definition  of  Directrices 

PFi 

6.  The  Ratio  Property,  -pg  =  e 

262 

7.  The  Latus  Rectum  =  — 


1.  Definition  of  Foci  and  Focal 
Radii 

2.  a^  +  b-'  =  c2 

3.  ra  —  ri  =  2a 

4.  Eccentricity,  e  =  -  =  sec  /3 

5.  Definition  of  Directrices 

PFi 

6.  The  Ratio  Property,  p^  =  e 

2b^ 

7.  The  Latus  Rectum  =  — 


410       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§234 

Exercises 

1.  Find  the  eccentricity  and  axes  of  sV^  —  y'/^^  =  1- 

2.  Find  the  eccentricity  and  latus  rectum  of  the  hyperbola  con- 
jugate to  the  hyperbola  of  the  preceding  exercise. 

3.  A  hyperbola  has  a  transverse  axis  equal  to  14  units  and  its 
asymptotes  make  an  angle  of  30°  with  the  Z-axis.  Find  the  equation 
of  the  hyperbola. 

4.  Find  the  latus  rectum  and  locate  the  foci  and  asymptotes  of 
4i2  -  362/2  =  144. 

6.  Locate  the  directrices  of  the  hyperbola  of  the  preceding  exercise. 

6.  In  Fig.  169  show  that  rz  =  GK'  and  ri  =  GI  and  hence  that 
rj  —  ri  =  IK'  or  2a. 

7.  Find  the  equation  of  the  hyperbola  having  latus  rectum  4/3 
and  a  =  26. 

8.  The  eccentricity  of  a  hyperbola  is  3/2  and  its  directrices  are  the 
lines  X  =  2  and  x  =  —  2.  Write  the  equation  and  draw  the  curve 
with  its  asymptotes,  a-circle,  6-circle,  and  foci. 

9.  Find  the  eccentricity  and  axes  of  3x^  —  5y'  =  —  45. 

10.  Find  the  eccentricity  of  the  rectangular  hyperbola. 

11.  Describe  the  shape  of  a  hyperbola  whose  eccentricity  is  nearly 
unity.  Describe  the  form  of  a  hyperbola  if  the  eccentricity  is  very 
Large. 

12.  Describe  the  hyperbola  if  b/a  =  2,  but  a  very  small. 

13.  Write  the  equation  of  the  hyperbola  if  (1)  c  =  6,  o  =  3;  (2) 
c  =  .25,  o  =  24;  (3)  c  =  17,  6  =  8.  / 

14.  Describe  the  locus  (,x  +  1)V7  -  iy  -  3)V5  =  1. 

15.  Find  the  equation  of  the  hyperbola  whose  center  is  at  the  origin 
and  whose  transverse  axis  coincides  with  the  X-axis  and  which  passes 
through  the  points  (4.5,  —  1),  (6,  8). 

234.  The  Polar  Equation  of  the  Ellipse  and  Hyperbola.     In 

mechanics  and  astronomy  the  polar  equations  of  the  ellipse  and 
hyperbola  are  often  required  with  the  pole  or  origin  at  the  right- 
hand  focus  in  the  case  of  the  ellipse  and  at  the  left-hand  focus  in 
the  case  of  the  hyperbola.  In  these  positions  the  radius  vector 
of  any  point  on  the  curve  will  increase  with  the  vectorial  angle 
when  B  <  180°.  To  obtain  the  polar  equation  of  the  ellipse  and 
hyperbola,  make  use  of  the  ratio  property  of  the  curves,  namely: 
That  the  locus  of  a  point  whose  distances  from  a  fixed  point  (called 
the  focus)  is  in  a  constant  ratio  e  to  its  distances  from  a  fixed 


§234] 


THE  CONIC  SECTIONS 


411 


line  (called  the  directrix),  is  an  ellipse  if  e  <  1  or  a  hyperbola 
if  e  >  1.  In  Fig.  170  let  F  be  the  fixed  point,  or  focus,  IK  the 
fixed  line,  or  directrix,  P  the  moving  point,  and  FL  =  I  the  semi- 
latus  rectum.  Then  the  problem  is  to  find  the  polar  equation 
under  the  condition 

pg-e  (1) 

If  e  is  understood  to  be  unrestricted  in  value,  the  work  and 
the  result  will  apply  equally  well  either  to  the  ellipse  or  to  the 
hyperbola. 


Fig.  170. — Polar  equation  of  a  conic. 

When  the  point  P  occupies  the  position  L,  Fig.  170,  we  have 
PF  =  I  and  PH  =  FN,  whence  from  (1) 

(2) 


FN  =  ^-. 
e 


Take  the  origin  of  polar  coordinates  at  F,  and  also  take  FP  =  p 
and  the  angle  AFP  =  6.    Then 

PH  =  FN  -  FD  (3) 

FD  =  p  cos  e.  (4) 


412       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§234 

Hence  from  (2),  (3),  and  (4) 

PH  =  -  -  p  cos  e.  (5) 

e 

Substituting  these  values  of  FP  and  PH  in  (1),  clearing  of  frac- 
tions and  solving  for  p,  we  obtain 

"  =  I  +  e  cos  e  ^^^ 

which  is  the  equation  required. 

When  e  <  1,  (6)  is  the  equation  of  an  ellipse  with  pole  at  the 
right-hand  focus.  When  e  >  1,  (6)  is  the  equation  of  a  hyperbola 
with  the  pole  at  the  left-hand  focus.  In  both  cases  the  origin  has 
been  so  selected  that  p  increases  as  d  increases. 

Note:  Calling  FN  (Fig.  170)  =  n,  equation  (1)  above  may  be 
written  in  rectangular  coordinates 

^^'  +  y'  =e,  (7) 

n  —  X 

x'  +  y^  =  e\n  -  x)'  ^°' 

which  may  be  reduced  to  the  form 

/  we'   \  2  y'      _       e'n' 

r  "•■  1  -  eV     "^  1  -  e'       (1    -e')^'  ^^' 

By  §§86  and  90  this  represents  an  ellipse  if  e  <  1  or  a  hyperbola 
if  e  >  1.  Thus  starting  with  the  ratio  definition  (7)  we  have  proved 
that  the  curve  is  an  ellipse  or  a  hyperbola;  that  is,  we  have  proved 
the  statements  in  italics  at  end  of  §§229  and  232. 

Exercises 

n 

1.  Graph  on  polar  paper,  form  M3,  the  curve  p  =  j—r-^ i 

for  e  =  2,  e  =  1/2,  and  e  =  1. 

It  will  be  sufficient  in  graphing  to  use  9  =  0°,  30°,  60°,  90°,  120°, 
160°,  180°,  210°,  .        .,  360°. 

2.  Write  the  polar  equation  of  an  ellipse  whose  semi-latus  rectum 
is  6  feet  and  whose  eccentricity  is  1/3. 

3.  Write  the  polar  equation  of  an  ellipse  whose  semi-axes  are  5  and  3. 

4.  Discuss  equation  (6)  for  the  case  e  =  0. 


§235] 


THE  CONIC  SECTIONS 


413 


6.  Write  the  polar  equation  of  a  hyperbola  if  the  eccentricity  be 
-\/2  and  the  distance  from  focus  to  vertex  be  4. 
6.  Write  the  polar  equation  of  the  asymptotes  of 


4  +  5  cos 

e 

9 

-and 

p  = 

9 

e' 

4+5  cos 
I 

4 

—  5  cos 

in 

which 

a 

is 

1  +e  cos 

(9- 

«)' 

7.  Compare  the  curves  p  = 

8.  Discuss  the  equation  p  = 

constant. 

235.  Ratio  Definition  of  the  Parabola.    Among  the  curves  of  the 
parabolic  type  previously  discussed,  the  one  whose  equation  is  of 


Fig.  171. — Properties  of  the  parabola  y^  =  ipx. 


the  second  degree  is  of  paramount  importance.  On  that  account 
when  the  term  parabola  is  used  without  qualification,  it  is  under- 
stood that  the  curve  is  the  parabola  of  the  second  degree,  whose 
equation  may  be  written,  y^  =  ax  or  x^  =  ay. 

We  shall  prove :  The  locus  of  a  'point  whose  distance  from  a  fixed 
point  is  always  equal  to  its  distance  from  a  fixed  line,  is  a  parabola. 
In  Fig.  171,  let  F  be  the  fixed  point  and  HK  the  fixed  line.  Take 
the  origin  at  A  half  way  between  F  and  HK.  Let  P  be  any  point 
satisfying  the  condition  PF  ==  PH.  Call  OD  =  x,  PD  =  y,  and 
represent  the  given  distance  FK  by  2p.  Then,  from  the  right 
triangle  PFD, 


414       ELEMENTARY  MATHEMATICAX  ANALYSIS       [§230 

ppi  =  2/2  +  FD^  (1) 

=  2/2  +  {x  -  OFY 
=  1/2  +  (a;  -  p)2. 

Since  PF  by  definition  equals  PH  or  a;  +  p,  we  have 

{x  +  vY  =  2/2  +  {x  -  v)\  (2) 

whence 

y^  =  4px,  (3) 

which  is  the  equation  of  the  parabola  in  terms  of  the  focal  distance, 
OF  or  p. 

The  double  ordinate  through  F  is  called  the  latus  rectum. 

The  semi-latus  rectum  can  be  obtained  from  (3)  by  placing 
X  =  p,  whence 

1  =  2p,  (4) 

where  /  is  the  semi-latus  rectum.    Hence  the  entire  latus  rectum  is 
4p,  or  the  coefficient  of  x  in  equation  (3). 

In  Fig.  171,  the  quadrilateral  FLIK  is  a  square  since  FL  and 
FK  are  each  equal  to  2p. 

236.  Polar  Equation  of  the  Parabola.  In  accordance  with  the 
ratio  definition  of  the  parabola,  its  polar  equation  is  found  at 
once  from  equation  (6),  §234,  by  putting  e  =  1.  Hence  the  polar 
equation  of  the  parabola  is 

''  =  7Tiosl"  (^) 

For  this  equation  we  may  make  the  following  table  of  values: 


e 

P 

0° 

1/2 

90° 

I 

180° 

00 

270° 

I 

This  shows  that  the  parabola  has  the  position  shown  in  Fig.  171. 
This  is  the  form  in  which  the  polar  equation  of  the  parabola  is 
used  in  mechanics  and  astronomy. 

237.  The  Conies.    It  is  now  obvious  that  a  single  definition 
can  be  given  that  will  include  the  ellipse,  hyperbola  and  parabola. 


§237]  THE  CONIC  SECTIONS  415 

These  curves  taken  together  are  called  the  conies.  The  definition 
fnay  be  worded:  A  conic  is  the  locus  of  a  point  whose  distances 
from  a  fixed  point  (called  the  focus)  and  a  fixed  line  (called  the 
directrix)  are  in  a  constant  ratio.  The  unity  between  the  three 
curves  was  shown  by  their  equation  in  polar  coordinates.  Moving 
the  ellipse  so  that  its  left  vertex  passes  through  the  origin,  as  in 
§85,  and  writing  the  hyperbola  with  the  origin  at  the  right  ver- 
tex (so  that  both  curves  pass  through  the  origin  in  a  comparable 
manner),  we  may  compare  each  with  the  parabola  as  follows: 

TheeUipse:  y^  =  2lx  -  (b^/a^)x''  (1) 

The  parabola:  j/'  =  2lx  (2) 

The  hyperbola:        y''  =  2lx  +  (b''/a')x''  (3) 

In  these  equations  I  stands  for  the  semi-latus  rectum  of  each 
of  the  curves.    These  equations  may  also  be  written 

1/2  =  2lx  -  (l/a)x'  (4) 

2/^  =  2lx  I      (5) 

2/2  =  2lx  +  (l/a)x''  (6) 

whence  it  is  seen  that  if  Z  be  kept  constant  while  a  be  increased 
without  limit,  the  ellipse  and  hyperbola  each  approach  the  parab- 
ola as  near  as  we  please.  Only  for  large  values  of  x,  if  a  be  large, 
is  there  a  material  difference  in  the  shapes  of  the  curves. 

Exercises 

1.  Write  the  equation  of  the  circle  in  the  form  (1)  above. 

2.  Write  the  equation  of  the  equilateral  hyperbola  in  the  form  (3)  ■ 
above. 

3.  Describe  the  curve 

^  I 

''     1  -I-  cos  (e  -  a)' 
where  a  is  a  constant. 

4.  In  Fig.  172  translate  the  curve  xy  =  Ihy  suitable  change  in  the 
equation  to  the  position  shown  by  the  dotted  curve,  if  the  translation 
of  each  point  is  unity. 


416      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§237 

6.  In  Pig.  173  translate  the  curve  j/*  =  4px  by  suitable  change  in 
the  equation  to  the  position  shown  by  the  dotted  curve,  if  the  distance 
each  point  is  moved  be  3p. 


Fig.  172. — A  hyperbola  translated  at  an  angle  of  45°  to  OX. 

6.  A  bridge  truss  has  the  form  of  a  circular  segment,  as  shown  in 
Fig.  174.     If  the  total  span  be  80  yards  and  the  altitude  BS  be  20 


Y 

0^ 

j^ 

0 

^-^^ 

X 

— —i. 

A  l»  Ci  10  Ca 


Fig.   173. — A  parabola  translated    Fig.  174. — Bridge  truss  in  form 
at  an  angle  of  60°  to  OX.  of  circular  segment. 

yards,  fmd  the  ordinates  CiDi,  C2D2,  etc.,  erected  at  uniform  intervals 
of  10  yards  along  the  chord  AAi. 

7.  A  bridge  truss  has  the  form  of  a  parabolic  segment,  as  shown  in 
Fig.  175.    The  span  AAi  is  24  yards  and  the  altitude  OB  is  10 


§238] 


THE  CONIC  SECTIONS 


417 
erected  at 


yards.    Find  the  length  of  the  ordinates  CD,  CiDi,  . 
untform  intervals  of  3  yards  along  the  Une  AAi- 

238.*  The  Conies  are  Conic  Sections.    The  curves  nowtnown  as 
the  conies  were  originally  studied  by  the  Greek  geometers  as  the  sec- 


X 

3        6 

9       12 

•^'^^ 

OoS 

\ 

\ 

Ai                               B 
Y 

Cs     Ci     C      A 

Fig.  175. — Bridge  truss  in  the  form  of  a  paraboUc  segment. 

tions  of  a  circular  cone  cut  by  a  plane.  It  can  be  shown  that  the  three 
classes  of  curves,  parabola,  eUipse,  and  hyperbola,  can  be  made 
respectively  by  cutting  any  circular  cone :  (1)  by  a  plane  parallel  to  an 


Fig.  176. — Section  of  a  circular  cone. 

element;  (2)  by  a  plane  cutting  opposite  elements  of  the  same  nappe  of 
the  cone;  (3)  by  a  plane  cutting  both  nappes  of  the  cone.  The  two 
nappes  of  a  conical  surface,  it  will  be  remembered,  are  the  two  portions 
of  the  surface  separated  by  the  apex. 

27 


418       ELEMENTARY  MATHEMATICAL  ANALYSLS       [§239 

In  Fig.  176,  let  the  plane  NDN'D',  caUed  the  cutting  plane,  cut  the 
lower  nappe  of  a  right  circular  cone  in  the  curve  VPV.  It  can  be 
proved  by  geometry  that  this  curve  is  an  ellipse.  The  foci  F  and  F 
are  the  points  at  which  the  two  inscribed  spheres  SFS'  and  RF'R'  are 
tangent  to  the  plane  ND'.  The  directrices  are  the  two  lines  ND 
and  N'D'  in  which  the  plane  ND'  cuts  the  two  planes  SHS'  and 
RKR'  produced. 

239.  Tangent  to  the  Parabola.  Let  us  investigate  the  condition 
that  the  line  y  =  mx  +  b  shaU  be  tangent  to  the  parabola  y^  = 
ipx.  First  find  the  points  of  intersection  of  these  loci  by  solving 
the  two  equations 

y    =  mx  +  6  (1) 

2/2  =  4pa;  (2) 

as  simultaneous  equations. 

Eliminating  y  by  substituting  the  value  of  y  from  (1)  in  (2), 

mV  +  2mbx  +  b'  -  ipx  =  0,  (3) 

or 

mV  +  2{mb  -  2p)x  +  b^  ==  0.  (4) 

Solving  for  x  (see  formula  for  quadratic.  Appendix  §309,  (2)). 

_  _  mb  —  2p        2\/p'  —  pmb  /gx 

m^  ~  m^ 

Therefore  there  are  in  general  two  values  of  x  or  two  points  of 
intersection  of  the  straight  line  and  the  parabola.  By  the  defini- 
tion of  a  tangent  to  a  curve  (§146)  this  line  becomes  a  tan- 
gent to  the  parabola  when  the  two  points  of  intersection  become 
a  single  point;  that  is,  when  the  expression  under  the  radical  in  (5) 
approaches  zero.     This  condition  requires  that 

p^  —  pmb  =  0, 
or 

b  =  p/m.  (6) 

Therefore  when  6  of  equation  (1)  has  this  value,  the  line  is  tangent 
to  the  parabola.    The  equation  of  the  tangent  line  is,  therefore 

y  =  mx  +  p/m.  (7) 

This  line  is  tangent  to  the  parabola  y^  =  ipx  for  all  values  of 


§240]  THE  CONIC  SECTIONS  419 

m.    Substituting  in  (5)  the  value  of  &  =  p/m,  we  may  find  the 

abscissa  of  the  point  of  tangency 

i 
xi  =  p/m'.  (8) 

Substituting  this  value  of  x  in  (7)  the  corresponding  ordinate  o' 
this  point  is  found  to  be 

yi  =  2p/m.  (9) 

240.  Properties  of  the  Parabola.  In  Fig.  171,  F  is  the  focus, 
HK  is  the  directrix,  PT  is  a  tangent  at  any  point  P  The 
perpendicular  PN  to  the  tangent  at  the  point  of  tangency  is 
called  the  normal  to  the  parabola.  The  projection  DT  of  the 
tangent  PT  on  the  X-axis  is  called  the  subtangent  and  the  pro- 
jection DN  of  the  normal  PN  on  the  X-axis  is  called  the  sub- 
normal. The  line  through  any  point  parallel  to  the  axis,  as  PR, 
is  known  as  a  diameter  of  the  parabola. 

(a)  The  subtangent  to  the  parabola  at  any  point  is  bisected  by 
the  vertex.  It  is  to  be  proved  that  OT  =  OD  for  all  positions  of  P 
Now  OD  is  the  abscissa  of  P,  which  has  been  found  to  be  p/m^. 
From  the  equation  of  the  tangent 

y  =  mx  +  p/m, 

the  intercept  OT  on  the  X-axis  is  found  by  putting  y  =  Q  and 
solving  for  x.     This  yields 

X  =  —  p/m}. 

This  is  numerically  the  same  as  OD,  hence  the  vertex  0  bisects 
DT. 

(6)  The  subnormal  to  the  parabola  at  any  point  is  constant  and 
eqval  to  the  semi-latus  rectum. 

The  angle  DPN  has  its  sides  mutually  perpendicular  to  the 
sides  of  the  angle  DTP,  hence  the  angles  are  equal.  Since  the 
tangent  of  the  angle  DTP  =  m,  therefore 

tangent  DPN  =  m. 

From  the  right  triangle  PDN, 

DN  =  PD  tan  DPN  =  PD  m 

=  i2p/m)m  =  2p. 


420       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§241 

Since  KF  also  equals  2p,  we  have 

DN  =  KF. 

(fi)  PFTH  is  a  rhombus.  To  prove  the  figure  PFTH  a  rhombus 
it  is  merely  necessary  to  show  that  FT  =  PH,  since  PF  =  PH 

FT  =  FO  +  OT 

PH  =  DK  =  DO  +  OK 
But 

OD  =  OT  and  OK  =  FO. 
Therefore 

FT  =  PH, 

and  1;he  figure  is  a  rhombus. 

It  follows  that  the  two  diagonals  of  the  rhombus  intersect  at 
right  angles  on  the  F-axis. 

(d)  The  normal  to  a  parabola  bisects  the  angle  between  the  focal 
radius  and  the  diameter  at  the  point.    We  are  to  show  that 

Z  NPF  =  Z  NPR. 
Since  FPHT  is  a  rhombus, 

Z  FPT  =  Z  TPH. 
But 

Z  TPH  =  Z  RPS, 

being  vertical  angles.    From  the  two  right  angles  NPT  and  NPS 
subtract  the  equal  angles  last  named.    Hence, 

Z  FPN  =  Z  NPR. 

It  is  because  of  this  property  of  the  parabola  that  the  reflectors 
of  locomotive  or  automobile  headlights  are  made  parabolic. 
The  rays  from"  a  source  of  light  at  F  are  reflected  in  lines  parallel 
to  the  axis,  so  that,  in  the  theoretical  case,  a  beam  of  light  is  sent 
out  in  parallel  lines,  or  in  a  beam  of  undiminishing  strength. 

241.  To  Draw  a  Parabolic  Arc.  One  of  the  best  ways  of  de- 
scribing a  parabolic  arc  is  by  drawing  a  large  number  of  tangent 
lines  by  the  principle  of  §240  (c).  Since  in  Fig.  171  the  tan- 
gent is  for  all  positions  perpendicular  to  the  focal  line  FH  at 
the  point  where  the  latter  crosses  OY,  it  is  merely  necessary  to 


§241] 


THE  CONIC  SECTIONS 


421 


draw  a  large  number  of  focal  lines,  as  in  Fig.  177,  and  erect 

perpendiculars  to  them  at  the  points  where  they  cross  the  F-axis. 

The  equations  of  the  tangent  lines  in  Fig.  177  are  of  the  form 


=  mx  +  p/m 


(1) 


Fig.  177. — Graphical  construction  of  a  parabolic  arc  "by  tangents." 

in  which  p  is  the  constant  given  by  the  equation  of  the  parabola, 
and  in  which  m  takes  on  in  succession  a  sequence  of  values  appro- 
priate to  the  various  tangent  lines  of  the  figure.  These  lines  are 
said  to  constitute  a  family  of  lines  and  to  envelop  the  curve  to 


422       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§242 

which  they  are  tangent.  The  curve  itself  is  called  the  envelope 
of  the  family  of  lines. 

The  curve  of  the  supporting  surface  of  an  aeroplane  as  well  as 
the  curve  of  the  propeller  blades  is  a  parabolic  arc.  The  curve  of 
the  cables  of  a  suspension  bridge  is  also  paraboUc. 

Exercises 

1.  Write  the  equation  of  the  parabola  which  the  family  y  =  mx 
+  7 /2m  envelops. 

2.  Draw  an  arc  of  a  parabola  if  p  =  3  inches. 

3.  At  what  point  is  ^  =  mx  +  Z/m  tangent  to  the  parabola  y'  = 
121? 

4.  At  what  point  isy  =  mx  +  ll/ira  tangent  to  y^  =  44x? 

6.  Draw  the  family  of  lines  y  =  mx  +  1/m  for  m  =  0.4,  m  =  0.6, 
m  =  0.8,  m  =  1,  m  =  2,  m  =  4,  m  —  8. 

242.  Tangent  to  the  Circle.  An  equation  of  a  tangent  hne  to 
a  circle  can  be  found,  as  in  the  case  of  the  parabola  above,  by 
finding,  the  points  of  intersection  of 

y  =  mx  +  b  (1) 

and 

x^  +  y^  =  a^  (2) 

and  then  imposing  the  condition  that  the  two  points  of  intersection 
shall  become  a  single  point.  The  value  of  6  that  satisfies  this 
condition  when  substituted  in  (1)  gives  the  equation  of  the  re- 
quired tangent.  It  is  easier,  however,  to  obtain  this  result  by  the 
following  method.  In  Fig.  178  let  the  straight  line  be  drawn 
tangent  to  the  circle  at  T.  Let  the  slope  of  this  line  be  m. 
Then  m  =  tan  ONT  =  tan  a,  if  a  be  the  direction  angle  of  the 
tangent  line.  The  intercept  b  of  t*he  line  on  the  F-aris  can  be  ex- 
pressed in  terms  of  a  and  a, 


b  =  a  sec  a  =  ay/l  -{-  m'^.  (3) 

Hence  the  equation  of  the  tangent  to  the  circle  is 

y  =  mx  +  a-\/i  +  m^. 

The  double  sign  is  written  in  order  to  include  in  a  single  equation 
the  two  tangents  of  given  slope  m,  as  illustrated  in  the  figure. 


§243] 


THE  CONIC  SECTIONS 


423 


Exercises 

1.  Find  the  equations  of  the  tangents  to  x'  +  y^  =  16  making  an 
angle  of  60°  with  the  X-axis. 

2.  Find  the  equations  of  the  tangents  to  x'  +  y^  =  25  making  an 
angle  of  45°  with  the  X-axis. 

3.  Find  the  equation  of  tangents  to  x^  +  y^  =  25  parallel  to  y  = 
3x  -  2. 

4.  Find  the  equation  of  tangents  to  x^  +  y'  =  16  perpendicular  to 
y  =  (l/2)x  +  3. 

5.  Find  the  equations  of  the  tangents  to  (a;  —  3)^  -|-  (?/  —  4)^  =  25 
whose  slope  is  3. 


FiG.  178. — The  equation  of  a  line  of  given  slope,  tangent  to  a  given 

circle. 

6.  Find  the  equation  of  the  tangent  to  the  circle  by  the  method 
of  §239. 

243.  Nonnal  Equation  of  Straight  Line.  The  normal  equation 
of  the  straight  line  was  obtained  in  polar  coordinates  in  §71. 
The  equation  was  written 

p  cos  {d  —  a)  =  a.  (1) 

In  this  equation  (p,  d)  are  the  polar  coordinates  of  any  point  on 
the  line,  a  is  the  distance  of  the  line  from  the  origin,  and  a  is  the 
direction  angle  of  a  perpendicular  to  the  line  from  the  origin. 
(See  Fig.  71.)     Expanding  cos  {6  —  a)  in  (1)  we  obtain 

f>  cos  6  cos  a  -J-  p  sin  5  sin  a  =  a.  (2) 


424       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§243 

But  for  any  value  of  p   and  d,  p  cos  B  =  x  and  p  sin  9  =  y. 
Hence  (2)  may  be  written  in  rectangular  coordinates 

X  cos  a  +  y  sin  a  =  a.  (3) 

This  also  is  called  the  normal  equation  of  the  straight  line. 
If  an  equation  of  any  line  be  given  in  the  form 

ax  +  by  =  c  (4) 

it  can  readily  be  "reduced  to  the  normal  form.     Dividing  this 
equation  through  by  s/a^  +  b^, 

"  b  c  .,, 

X  +       .  y  =       .  (5) 


Va'  +  b^  Va^  +  h^  VaT+h^' 

Now  aly/a?'  +  6^  and  b/-\/a^  +  h^  may  be  regarded  as  the  cosine 
and  sine,  respectively,  of  the  angle  formed  with  the  positive  Z-axis 
by  the  line  joining  the  origin  to  the  point  (a,  6).  Calling  this 
angle  a,  equation  (5)  may  be  written 

X  cos  a  +  2/  sin  o;  =  d,  (6) 

which  is  of  the  form  (3)  above.  Inasmuch  as  the  right-hand  side 
of  the  equation  in  the  normal  form  represents  the  distance  of  the 
line  from  the  origin,  it  is  best  to  keep  the  right-hand  Side  of  the 
equation  positive.  The  value  of  a  and  the  quadrant  in  which  it 
lies  is  then  determined  by  the  signs  of  cos  a  and  sin  a  on  the  left- 
hand  side  of  the  equation.  The  angle  a  may  have  any  value  from 
0°  to  360°. 

Illustrations: 

•  (1)  Put  the  equation  Zx  —  ^y  =  10  in  the  normal  form.     Here 
a2  +  62  =  3'  +  ( -  4)2  =25.     Dividing  by  5  we  obtain 

{Z/5)x  -  (4/5)2/  =  2. 

The  distance  of  this  line  from  the  origin  is  2.  The  angle  a  is  the  angle 
whose  cosine  is  3/5  and  whose  sine  is  —  4/5.  Therefore,  from  the 
tables,  a  =  306°  52'. 

(2)  Put  the  equation  3a;  —  4v  4-  20  =  0  in  the  normal  form.  Trans- 
posing and  dividing  by  —  1  to  make  the  right-hand  side  of  the  equa- 
tion positive,  we  obtain  —  3a;  -(-  4i/  =  20. 

Here  cos  a  =  -  3/5,  sin  a  =  4/5,  d  =  4.    Hence  a  =  126°  52'. 


§244]  THE  CONIC  SECTIONS  425 

(3)  What  is  the  distance  between  the  lines  (1)  and  (2)?  The  lines 
are  parallel  and  on  opposite  sides  of  the  origin.  Their  distance  apart 
is  therefore  2  +  4  or  6. 

(4)  Put  X  +  y  =  I  in  the  normal  form.  Here  VoM- b^  =  V'2. 
The  equation  becomes  i  -s/^x  +  i  ■\/2y  =  i  y/2.     a  =  45°,  d  =  i  \/2. 

Exercises 

1.  The  shortest  distance  from  the  origin  to  a  line  is  5  and  the  direc- 
tion angle  of  the  perpendicular  from  the  origin  to  the  line  is  30°- 
Write  the  equation  of  the  line. 

2.  The  perpendicular  from  the  origin  upon  a  straight  line  makes  an 
angle  of  135°  with  OX,  and  its  length  is  2v'2.  Find  the  equation  of 
the  line. 

3.  Write  the  equation  of  a  straight  line  in  the  normal  form  if  a  = 
60°  and  d  =  y/s. 

4.  Put  2\/3a;  +  2?/  =  32  in  the  normal  form  and  find  the 
numerical  values  of  a  and  d. 

6.  Put  2a;  —  2i/  =  1  in  the  normal  form  and  find  the  values  of  a 
and  d. 

6.  Find  the  equation  of  the  straight  line,  if  the  perpendicular  from 
the  origin  on  the  line,  makes  an  angle  of  46°  with  the  X-axis  and  its 
length  is  ■\/2. 

7.  Put  2  -ho  =  1  ii  the  normal  form. 

244.  To  Translate  Any  Locus  a  Given  Distance  in  a  Given  Direc- 
tion. To  move  any  locus  the  distance  d  to  the  right  we  sub- 
stitute (xi  —  d)  for  X  in  the  equation  of  the  locus.  To  move  the 
locus  the  distance  d  in  the  y  direction  we  substitute  (2/1  —  d)  for  y. 
To  move  any  locus  the  distance  d  in  the  direction  a  we  substitute 

(xi  —  d  cos  a)  for  x,  ,  , 

(2/1  —  d  sin  a)  for  y, 

which  must  give  the  desired  equation  of  the  new  locus.  It  is 
not  necessary  to  use  the  subscript  attached  to  the  new  coordinates 
if  the  distinction  between  the  new  and  old  coordinates  can  be 
kept  in  mind  without  this  device. 

The  circle  x^  +  y^  =  a^  moved  the  distance  d  in  the  direction 
a  becomes 

(x  —  d  cos  a)^  -i-  (y  —  d  sin  a)"  =•  a'^ 


426      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§245 

which  may  be  changed  to 

x^  —  2Ax,  cos  a  +  2/^  —  2^2/  sin  a  =  a?-  —  cP. 

245.  Distance  of  Any  Point  From  Any  Line.    Let  the  equation 
of  the  line  I,  Kg.  179,  in  the  normal  form  be 

X  cos  a  +  2/  sin  a  =  a,  (1) 

and  let  (a;i,  t/i)  be  any  point  P  in  the  plane.  (See  Fig.  179.) 
If  the  point  {xi,  yi)  is  on  the  opposite  side  of  the  line  froin  the 
origin,  the  line  can  be  moved  so  that  it  will  pass  through  the  point 
by  translating  it  the  proper  distance  in  the  direction  a.  Let 
the  unknown  amount  of  the  required  translation  be  represented 
by  d.  To  translate  the  line  the  amount  d  in  the  direction  a, 
we  must  substitute  for  x  and  y  the  values 

X  =  x'  —  d  cos  a  .  . 

y  =  y'  —  d  cos  a 
We  obtain 

(x'  —  d  cos  a)  COB  a  +  iy'  —  d  sin  a)  sin  a  =a.  (3) 

The  line  represented  by  this  equation  passes  through  the  point 
(xi,  2/i).  Substituting  these  coordinates  for  x'  and  y'  and  solving 
for  d,  we  have 

d  =  Xi  COS  a  +  yi  sin  a  —  a.  (4) 

This  is  the  distance  of  (xi,  yi)  from  the  given  line. 

If  the  given  point  is  on  the  same  side  of  the  line  as  the  origin, 
as  the  point  Pixi,  y^  Fig.  179,  then  the  given  line  must  be 
translated  the  distance  d  in  the  direction  (180°  +  a),  and  the  result 
is  the  same  as  (4)  above  except  all  signs  are  changed.  We  are  usu- 
ally interested  only  in  the  numerical  value  of  d,  so  that  formula 
(4)  may  be  used  for  all  cases.  When  the  value  of  d  comes  out 
negative  it  merely  means  that  the  given  point  is  on  the  same  side 
of  the  line  as  the  origin. 

Equation  (4)  may  be  interpreted  as  follows : 

To  find  the  distance  of  any  point  from  a  given  line,  put  the  equa- 
tion of  the  line  in  the  normal  form,  transpose  all  terms  to  the  left- 
hand  member  and  siibstitvte  the  coordinates  of  the  given  point  for  x 
and  y.  The  absolute  value  of  the  left-hand  member  is  the  distance 
of  P  from  the  line. 


§245] 


THE  CONIC  SECTIONS 


427 


The  above  facts  may  be  stated  in  an  interesting  form  as  follows: 
Let  any  line  be 

X  cos  a  +  2/  sin  a  —  a  =  0. 

If  the  coordinates  of  any  point  on  this  line  be  substituted  in  this 
equation,  the  left-hand  member  reduces  to  zero.  If  the  coordi- 
nates of  any  point  not  on  the  line  be  substituted  for  x  and  y  in  the 
equation,  the  left-hand  member  of  the  equation  does  not  reduce 
to  zero,  but  becomes  negative  if  the  given  point  is  on  the  origin 
side  of  the  line  and  positive  if  the  given  point  is  on  the  non- 
origin  side  of  the  line .    The  absolute  value  of  the  left-hand  member 


\ 


\'^ 


Y 

\ 

\ 

\ 

2 

\ 

\ 

\ 

\ 

\ 

cZ^APi(<Kny,) 

\ 
\ 
\ 

\ 

4^\ 

\ 

k\  \ 

\ 

. 

\,-y\             ^ 

\ 

> 

^ 

a    \           \ 

P.ik^, 

!/=)     >             \ 

\ 

<^  \          \ 

0 

/- 

/ 

\                          \                       -A    \ 

\                          \                             \_ 

Fig.  179. — Distance  of  any  point  from  a  given  Une. 

in  each  case  gives  the  distance  of  the  given  point  from  the  line. 
Thus  every  line  may  be  said  to  have  a  "positive  side"  and  a 
"negative  side."  The  "negative  side"  is  the  side  toward  the 
origin. 

Illtjstbation  1.     Find  the  distance  of  (—1,  4)  from  the  line  3a;  — 
41/  =  10. 
Transpose  and  put  left-hand  member  in  normal  form 


1^  —  iV 


0. 


428       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§246 

Subatitute  —  1  for  a;  and  4  for  y.    The  left  member  is  now  the  value 
of  d,  BO  that 

The  result  is  negative,  so  that  ( —  1,  4)  is  on  the  same  side  of  the  line  as 
the  origin. 
iLLtrsTRATioN  2.    Find  the  distance  of  (2,  —  4)  from 

x-2      y+3 
4  7 

Clear  of  fractions  and  simplify, 

7a;  -  4?/  -  26  =  0. 
Put  in  normal  form, 

iz^  -  i\y  -  u  =  0. 

Substitute  2  for  x  and  —  4  for  y, 

The  point  is  on  the  non-origin  side  of  the  given  line,  and  irV  of  one  imit 
from  it. 

Exercises 

1.  Find  the  distance  of  the  point  (4,  5)  from  the  line  3x  +  iy  =  10. 

2.  Find  the  distance  from  the  origin  to  the  line  x/3  —  2//4  =  1. 

3.  Find  the  distance  from  (-3,  -  4)  to  12(a;  +  6)  =  6(2/  -  2). 

4.  Find  the  distance  from  (3,  4)  to  the  line  x/3  —  y/4  =  1. 

5.  Find  the  distance  between  the  parallel  lines  y  =  2x  -\-  3,  and 
y  =  2x  +5. 

6.  Find  the  distance  between  y  =  2x  —  3,  y  =  2x  +  5. 

7.  Find  the  distance  from  (0,  3)  to  4a;  —  3y  =  12. 

8.  Find  the  distance  from  (0, 1)  to  a;  +  2  —  2?/  =0. 

246.  Tangent  to  a  Circle  at  a  Given  Point.  The  equation  of 
a  line  having  a  given  slope  m  and  tangent  to  a  given  circle  with 
center  at  the  origin,  was  given  in  §242.  We  shall  now  find  the 
equation  of  the  line  that  is  tangent  to  the  circle  at  a  given  point 
(xo,  2/o). 
The  line, 

a  ='p  cos  {6  —  a),  (1) 

or  its  equivalent, 

a;  cos  a  +  2/  sin  a  =  a,  (2) 


§247] 


THE  CONIC  SECTIONS 


429 


is  tangent  to  the  circle  of  radius  a,  and  the  point  of  tangency  is  at 
the  end  of  the  diameter  whose  direction  angle  is  a.  The  point  of 
tangency  is  therefore  (o  cos  a,  a  sin  a).  Hence,  multiplying  (2) 
through  by  a,  we  obtain 

x{a  cos  a)  +  y(a  sin  a)  =  a^,  (3) 

or 

xox  +  yoy  =  a.K  (4) 

This  is  the  equation  of  the  hne  tangent  to  the  circle  of  radius  a 
at  the  point  (xq,  ya). 
Thus  3a;  +  4?/  =  25  is  tangent  to  a;^  +  y"^-  =  25  at  (3,  4). 

247.  Tangent  to  the  Ellipse  at  a  Given  Point.    It  is  easy  to 
draw  the  tangent  to  the  ellipse  at  any  desired  point.     In  Fig.  180, 


Fig.  180. — Tangent  to  the  ellipse  at  a  given  point. 


let  Po  be  the  point  at  which  a  tangent  is  desired.  Draw  the 
major  circle,  and  let  Pj  of  the  circle  be  a  point  on  the  same  ordinate 
asPo.  Draw  a  tangent  to  the  circle  at  Pi  and  let  it  meet  the 
Z-axis  at  T.  Then  when  the  circle  is  projected  to  form  the 
ellipse,  the  straight  line  PiT  is  projected  to  make  the  tangent  to 
the  ellipse.  Since  T  when  projected  remains  the  same  point  and 
since  Po  is  the  projection  of  Pi,  the  line  through  Po  and  T  is  the 
required  tangent  to  the  ellipse. 


430      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§248 

The  equation  of  the  tangent  PoT  is  alao  readily  found.  The 
equation  of  PiT  is 

xxo  +  yyo   =  ffl^  (1) 

To  project  his  into  the  line  Po^  it  is  merely  necessary  to  multiply 
the  ordinates  y  and  yo'  by  b/a;  that  is,  to  substitute  y  =  ay/b  and 
2/o'  =  ayo/b.    Whence  (1)  becomes 

xox  +  a^yoy/b^  =  a^  (2) 

or,  dividing  by  a^, 

xox/a"  +  joj/b^  =  I  (3) 

which  is  the  tangent  to 

a;2/a2  -I-  2^2/62   =   1 

at  the  point  (xo,  yo)- 

Exercises 

1.  Find  the  equations  of  the  tangents  to  the  ellipse  whose  semi-axes 
are  4  and  3  at  the  points  for  which  x  =  2. 

2.  Find  the  equations  of  the  tangents  to  x'/16  +  y^/9  =  1  at  the 
ends  of  the  left-hand  latus  rectum. 

3.  Required  the  tangents  to  x'/9  +  y'/i  =  1,  making  an  angle  of 
45°  with  the  X-axis.  [Solve  y  =  x  +  b  and  x'/9  +  y"/4  =  1  as  in 
§239.] 

4.  Find  the  equatioiis  of  the  tangents  to  sr^/lOO  -I-  y'/25  =  1  at 
the  points  where  y  =  3. 

6.  Find  the  equations  of  the  tangents  to  x'/S6  +  y'/16  =  1  at  the 
^ints  where  x  =  y. 

248.  The  Tangent,  Normal,  and  Focal  Radii  of  the  Ellipse.    In 

the  right  triangle  PiOT,  Fig.  180,  the  side  PiO  is  a  mean  propor- 
tional between  the  entire  hjrpotenuse  OT  and  the  adjacent 
segment  OD.    That  is 

a^  =  xoX  OT,  or  OT  =  a^/xo 
Then  FiT  =  OT  -  OFi  =  OT  -  ae 

=  a'^/xo  —  ae 
Wkewis^  FtT  =  OT  +  OFi 

5=  aVaio  ■+-  ae 


§249]  THE  CONIC  SECTIONS  431 

Therefore  FrT/FiT  =  {a^xa  -  ae)/{ayxa  +  ae) 

=  (a  —  exo)/{a  +  exo) 

But  by  §230  this  last  ratio  is  equal  to  r^/r^.  Therefore  we 
may  write  FiT/F^T  =  P^F./PoF,. 

Hence  T,  which  divides  the  base  FiFi  of  the  triangle  PoFaFi 
externally  at  T  in  the  ratio  of  the  two  sides  PF^  and  PFi  of  the 
triangle,  lies  on  the  bisector  of  the  external  angle  FiPoQ  of  the 
triangle  FJPoFi.     This'  proves  the  important  theorem: 

The  tangent  to  the  ellipse  bisects  the  external  angle  between  the 
focal  radii  at  the  point. 

This  theorem  provides  a  second  method  of  constructing  a 
tangent  at  a  given  point  of  an  elUpse,  often  more  convenient 
than  that  of  §247,  since  the  method  of  §247  often  runs  the 
construction  off  of  the  paper. 

The  normal  PoN,  being  perpendicular  to  the  tangent,  must 
bisect  the  internal  angle  F^PoFi  between  the  focal  radii  F^o  and 
F,Po. 

Since  the  angle  of  reflection  equals  the  angle  of  incidence  for 
light,  sound,  and  other  wave  motions,  a  source  of  light  or  sound  at 
Fi  is  "brought  to  a  focus"  again  atF^,  because  of  the  fact  that  the 
normal  to  the  ellipse  bisects  the  angle  between  the  focal  radii. 

249.  Additional  Equations  of  the  Straight  Liae.^  The  equations 
of  the  straight  line  in  the  slope  form 

y  =  mx  +  b  (1) 

and  in  the  normal  forms 

p  cos  {d—  a)  =  a  (2) 

a;  cos  a  +  2/  sin  a  =  a  (3) 

and  the  general  form 

ax  +  by  +  c  =  0  (4) 

have  already  been  used.  Two  constants  and  only  two  are  neces- 
sary for  each  of  these  equations.  The  constants  in  the  first 
equation  are  m  and  6;  in  the  second  and  third,  a  and  a;  in  the 
fourth  a/c  and  b/c,  or  any  two  of  the  ratios  that  result  from  divid- 
ing through  by  one  of  the  coefficients.  Equation  (4)  appears  to 
contain  three  constants,  but  it  is  only  the  relative  size  of  these  that 

>  See  §17. 


432       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§249 

determines  the  particular  line  represented  by  the  equation,  since 
the  line  would  remain  the  same  when  the  equation  is  multiplied 
or  divided  through  by  any  constant  (not  zero). 

These  facts  are  usually  summarized  by  the  statement  that  two 
conditions  are  necessary  and  sufficient  to  determine  a  straight 
line.  The  number  of  ways  in  which  these  conditions  may  be  given 
is,  of  course,  unhmited.  Thus  a  straight  line  is  determined  if  we 
say,  for  example,  that  the  line  passes  through  the  vertex  of  an 
angle  and  bisects  that  angle,  or  if  we  say  that  the  line  passes 
through  the  center  of  a  circle  and  is  parallel  to  another  line,  or  if 
we  say  that  the  straight  Une  is  tangent  to  two  given  circles,  etc. 
An  important  case  is  that  in  which  the  line  is  determined  by  the 
requirement  that  it  pass  through  a  given  point  in  a  given  direc- 
tion. The  equation  of  the  line  adapted  to  this  case  is  readily 
found.  Let  the  given  point  be  {xi,  yi).  The  line  through  the 
origin  with  the  required  slope  is 

y  =  mx. 

Translate  this  line  so  that  it  passes  through  (xi,  yi)  and  we  have 

y  -  yi  =  m(x  -  xi).  (5) 

Another  way  of  obtaining  the  same  result  is :  Substitute  the 
coordinates  (.Xi,  yi)  in  (1) 

2/1  =  mxi  +  6.  (6) 

Subtract  the  members  of  this  from  (1)  above,  so  as  to  eliminate 
b.     There  results 

y  -  yi  =  m{x  —  a;i).  (7) 

This  is  the  required  equation.  The  given  point  is  (a;i,  2/1)  and  the 
direction  of  the  line  through  that  point  is  given  by  the  slope  m. 
Another  important  case  is  that  in  which  the  straight  line  is 
determined  by  requiring  it  to  pass  through  two  given  points. 
Let  the  second  of  the  given  points  be  (012,  yi).  Substitute  these 
coordinates  in  (5) 

2/2  -  2/1  =  »»(»2  -  a;i).  (8) 

To  eliminate  ?n,  divide  the  members  of  (7)  by  the  members  of  (8) 

2/  -  2/1         X  —  xi 


2/2  -  2/1       2:2  -  X\ 


(9) 


§250]  THE  CONIC  SECTIONS  433 

or,  as  it  is  usually  written 

L^Zli  ^Yl^zll^;  (10) 

X  —  Xi         X2  —  Xi 

0 

This  is  the  equation  of  a  line  passing  through  two  given  points. 
Since  (10)  may  be  looked  upon  as  a  proportion,  the  equation  may 
be  written  in  a  variety  of  forms. 

250.  The  Circle  Through  Three  Given  Points.  In  general,  the 
equation  of  a  circle  can  be  found  (when  three  pojnts  are  given. 
Either  of  the  general  equations  of  the  circle 

{x  -  hY  +{y  -  kY  =  a\  (1) 

or 

a;"  +  2/^  +  2gx  +  2/2/  +  c  =  0  (2) 

contains  three  unknown  constants,  so  that  in  general  three  con- 
ditions may  be  imposed  upon  them.  It  is  best  to  illustrate  the 
general  method  by  a  particular  example.  Let  the  three  given 
points  be  (—  1,  3),  (0,  2),  and  (5,  0).  Then  since  the  coordinates 
of  these  points  must  satisfy  the  equation  of  the  circle,  we  obtain 
from  (2)  above 

1  +  9  -  2ff  +  6/  +  c  =  0,  (3) 

4  +  4/  +  c  =  0,  (4) 

25  +\0g        +  c  =  0.  (5) 

Eliminating  c  from  (3)  and  (4)  and  from  (4)  and  (5),  we  obtain 

6  -  2sr^  +  2/  =  0, 

21  +  10^  -  4/  =  0. 
Eliminating  / 

?  =  -  5i 
Whence 

/  =  -  8i 
and 

a  =  30. 

So  the  equation  of  the  circle  is  ■ 

a;2  4-  yi   _    lla;    _    ny   +   30    =   0. 

28 


434      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§251 

Exercises 

1.  Knd  the  equation  of  the  line  passing  through  (2,  3)  with  slope 
2/3. 

2.  Find  the  equation  of  the  line  passing  through  (2,  3),  and  (3,  5). 

3.  Find  the  line  passing  through  (2,  —  1)  making  an  angle  whose 
tangent  is  2  with  the  Z-axis. 

4.  Find  the  line  through  (2,  3)  parallel  to  2/  =  7a;  +  11. 

5.  A  line  passes  through  (—1,  —  3)  and  is  perpendicular  to 
y  —  2x  —'  3.     Find  its  equation. 

6.  Find  the  line  passing  through  (—  2,  3),  and  (—3,-1). 

7.  Find  the  equation  of  the  line  which  passes  through  (—  1,  —3), 
and  (-2,4). 

8.  Find  the  slope  of  the  line  that  passes  through  ( —  1,  6 ),  and 
(-2,8). 

9.  Find  the  equation  of  the  line  passing  through  the  left  focus  and 
the  upper  end  of  the  right  latus  rectum  of  a;2/2S  +  y'/9  =  1. 

10.  Find  the  equation  of  the  circle  passing  through  (2,  8)„  (5,  7), 
and  (6,  6) . 

11.  Find  the  equation  of  the  circle  which  passes  through  (1,  2), 
(-  2,  3),  and  (-  1,  -  1). 

12.  Find  the  equation  of  the  parabola  in  the  form  y^  =  ipx  which 
passes  through  the  point  (2,  4). 

251.  Change  from  Polar  to  Rectangular  Coordinates.    The 

relations  between  x,  y  of  the  Cartesian  system  and  p,  6  of  the 
polar  system  have  already  been  explained  and  use  made  of 
them.    The  relations  are  here  brought  together  for  reference : 

X  =  p  cos  8  (1) 

y  =  p  sin  0.  (2) 

By  these  we  may  pass  from  the  Cartesian  equation  of  any  locus 
to  the  equivalent  polar  equation  of  that  locus.  Dividing  (2) 
by  (1)  and  also  squaring  and  adding,  we  obtain: 

e  =  tan-i  y/x  (3) 

P  =  Vx^  +  y^  (4) 

These  may  be  used  to  convert  any  polar  equation  into  the  Carte- 
sian equivalent. 

262.  Rotation  of  Any  Locus.  It  has  already  been  explained 
that  any  locus  can  be  rotated  tlirough  an  angle  a  by  substituting 


§252] 


THE  CONIC  SECTIONS 


435 


(Ot  —  a)  for  d  in  the  polar  equation  of  the  locus.  It  remains  to 
determine  the  substitutions  for  x  and  y  which  will  bring  about 
the  rotation  of  a  locus  in  rectangular  coordinates.  Let  us  consider 
any  point  P  of  a  locus  before  and  after  rotation  through  the  given 
angle  a.  Call  the  coordinates  of  the  point  before  rotation 
(x,  y)  in  rectangular  coordinates  and  (p,  6)  in  polar  coordinates.' 
Then,  from  (1)  and  (2),  §251, 

X  =  p  cos  6  (1) 

t/  =  P  sin  6.  (2) 

Call  the  coordinates  of  the  point  after  rotation  (xi,  yi)  and 

(Pi,  ^i),  but  note  that  the  value  of  p  is 

unchanged  by  the  rotation.     Then  for  p  (Pi,  e,)  or 

the  point  P',  Fig.  181,  we  may  write         ]         /\  (X1.V1) 

Xi  =  p  cos  61  (3) 

2/1  =  p  sin  01.  (4) 

Since,  however,  the  rotation  requires  — 

that  Fig-  181 

6  =  Oi  -  ot  (5) 

equations  (1)  and  (2)  become 

X  =  p  cos  (di  -^  a)  =  p  cos  di  cos  a  +  p  sin  81  sin  a      (6) 

2/  =  p  sin  (di  —  a)  =  p  sin  61  cos  a  —  p  cos  0i  sin  a.      (7) 

But,  from  (3)  and  (4),  p  cos  di  and  p  sin  di  are  the  new  values  of 

X  and  y;  hence,  substituting  in  (6)  and  (7)  from  (3)  and  (4)  we 

obtain 

X  =  xi  cos  Qi  +  yi  sin  a  (8) 

y  '=  yi  cos  a  —  Xi  sin  a.  (9) 

Hence,  if  the  equatiofi  of  any  locus  is  given  in  rectangular  co- 
ordinates, it  is  rotated  through  the  positive  angle  a  by  the  sub- 
stitutions 

X  cos  a  +  y  sin  a  for  X 

y  cos  Q!  —  X  sin  a  for  y,  (10) 

in  which  it  is  permissible  to  drop  the  subscripts,  if  the  context 
shows  in  each  case  whether  we  are^  dealing  with  the  old  x  and  y 
or  with  the  new  x  and  y. 


PiP.e)  or 


-Rotation  of 
any  locus. 


436       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§252 

If  the  required  rotation  is  clockwise,  or  negative,  we  must 
replace  a  by  (—  a)  in  aU  of  the  above  equations. 

Whenever  comenient,  the  eqvation  of  a  curve  should  he  taken  in 
the  -polar  form  if  it  is  required  to  rotate  the  locus. 

Important  Facts:  The  following  facts  should  be  remembered 
by  the  student: 

(1)  To  rotate  a  curve  through  90°,  change  x  to  y  and  y  to  {  —  x). 

(2)  Rotation  through  any  angle  leaves  the  expression  x^  +  y^ 
(fir  any  function  of  it)  unchanged.  This  is  obvious  since  the  circle 
x^  +  y^  =  a^  is  not  changed  by  rotation  about  (0,  0). 


Fig.  182. — Effect  of  rotation  on  the  special  forms  x^  +  y^,  2xy,  and 

x^  -  2/2. 

(3)  Rotation  through  +  45°  changes  2xy  to  y^  —  x^. 
Rotation  through  —  45°  changes  2xy  to  x^ ,—  y^. 

(4)  Rotation  through  +  45°  changes  x^  —  y^  to  2xy. 
Rotation  through  —  45°  changes  x'^  —  y^  to  —  2xy. 

Statements  (3)  and  (4)  follow  at  once  from  consideration  of  the 
equations 

i;2  -  2/2  =  a^  (1) 

a'  (2) 

a'  <3) 

a^  (4) 


2xy 

yi    _    j;Z 

-  2xy 


§253]  THE  CONIC  SECTIONS  437 

of  the  four  hyperbolas  bearing  corresponding  numbers  (I),  (2), 
(3),  (4)  in  Fig.  182.  The  proper  substitution  in  any  case  can 
be  remembered  by  thinking  of  the  four  hyperbolas  of  this  figure. 
(5)  The  degree  of  an  equation  of  a  locus  cannot  he  changed  by 
a  rotation.  This  follows  at  once  from  the  fact  that  the  equations 
of  transformation  (8)  and  (9)  are  linear. 

Exercises 

In  order  to  shorten,  the  work,  use  statements  (1)  to  (4)  whenever 
possible. 

1.  Turn  the  locus  a*  —  j/'  =  4  through  45°. 

2.  Turn  x'  +  y'  =  a'  through  79°.     Turn  ixy  =  1  through  45°. 

3.  Turn  x  cos  a  -\-  y  sin  a  =  a  through  an  angle  /3.  (Since  this 
locus  is  well  known  in  the  polar  form,  transformation  formulas  (6)  and 
(7)  above  need  not  be  used.) 

4.  Rotate  x'  -  y^  =  1  through  90°. 

5.  Rotate  s"  —  j/^  =  a'  through  —  45°. 

6.  Rotate  x'  -  y'  =  1  through  30°. 

7.  Rotate  x^  -  y"  =  4:  through  60°. 

8.  Rotate  ixy  =  1  through  30°. 

9.  Rotate  x'  +  2y'  =  1  through  45°. 

10.  Change  the  equation  (x  —  a)'  +  (y  —  b)'  =  rHo  the  polar  form. 

11.  Change  p  cos  29  =  2a,  one  of  a  class  of  curves  known  as  Cote's 
spirals,  to  the  Cartesian  form. 

12.  Write  the  equation  of  the  lemniscate  in  the  polar  form. 

13.  Show  that  p'  —  2p/oicos  (9  —  9i)  +  pi"  =  a'is  the  polar  equation 
of  a  circle  with  center  at  (pi,  9i)  and  of  radius  a. 

14.  Write  the  Cartesian  equation  of  the  locus  p"  =  16  sin  29. 

15.  Turn  p^  =  8  sin  29  through  an  angle  of  45°. 

16.  Rotate  x^  -  2y^  =  1  through  90°. 

17.  Rotate  {x^  +  y^)^  +  {x'  -  y')^  =  1  through  45°. 

18.  Rotate  log  {x'  +  y^)  =  tan  {x^  -  y^)  through  45°. 

19.  Rotate  x^  -&xy  -\-y''=\  through  45°. 

20.  Rotate  x^  +  y^^  =  a}^  through  45°.    Show  that  the  result  is 
the  parabola  y  =  x^la  +  o/2,  and  sketch  the  curves. 

253.  Ellipse  with  Major  Axis  at  45°  to  the  QX  Axis.    The 

ellipse  frequently  arises  in  applied  science  as  the  resultant  of  the. 
projection  of  the  motion  of  two  points  moving  uniformly  on  two 


438      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§253 

circles,  as  has  already  been  explained  in  §186.    Thus  the  para- 
metric equations  « 

X  =  a  cos  t  (1) 


2/  =  6  sin  t, 


(2) 


define  an  ellipse  which  may  be  considered  the  resultant  of  two 
S.H.M.  in  quadrature.    We  shall  prove  that  the  equations 


a;  =  o  cos  t 

y  =  asm  (t  +  a), 


(3) 
(4) 


define  an  ellipse,  with  major  axis  making  an  angle  of  45°  with  OX. 
The  graph  is  readily  constructed  as  in  Fig.  183.    The  Car- 
tesian equation  of  the  curve  is  found  by  eliminating  t  between 


/ 

/ 

^ 

Y 
B 

|\ 

Ic 

— 

■P' 

\^     "^ 

y 

¥    \y 

/ 

^' 

A 

X' 

/ 

.^^ 

■^ 

1  y\-    \x 

/ 

, 

-^ 

O 

/ 

A 

T  3 

.■'• 

/ 

1 

-^ 

y 

/ 

c 

^ 

^ 

r' 

Fig.  183. — The  ellipse  x  =  a  cos  t,y  —  a  sin  (<  +  a). 

(3)  and  (4).    Expanding  the  sin  {t  +  «)  in  (4)  and  substituting 
from  (3)  we  obtain 


(5) 


y  =  X  sin.a  +  -y/a^  —  x"^  cos  a. 
Transposing  and  squarimg 

a;^  —  2xy  sin  a  +  j/''  =  a^  cos'  a.  (6) 

By  §252  rotate  the  curve  through  an  angle  of  (—  46'').     We 


§254]  THE  CONIC  SECTIONS  439 

know  that  (x*  +  y^)  is  unchanged  and  that  2xy  is  to  be  replaced 
by  (a;2  —  y^).    Therefore  (6)  becomes 

x^l  —  sin  a)  +  y^{l  +  sin  a)  =  a^  cos^a.  (7) 

Replacing  cos^  a  by  1  —  sin^  a,  and  dividing  through  by  the 
right-hand  member,  we  obtain 

a\l  +  sin  a)  "^  a^l  -  sin  a)  "  ^'  ^^^ 

which  may  be  written 

—-—  +  -^,=  1,  (9) 


2a2  cos^  I      20^  sin''  |^ 

where  /8  is  the  complement  of  a.  Equation  (8)  or  (9)  proves 
that  the  locus  is  an  ellipse.  It  is  any  ellipse,  since  by  properly 
choosing  a  and  a  the  denominators  in  (8)  can  be  given  any  desired 
values.  Hence  the  pair  of  parametric  equations  (3)  and  (4),  or 
the  Cartesian  equation  (5)  represents  an  ellipse  with  its  major  axis 
inclined  +  45°  to  the  X-axis. 

254.  She.ar  of  the  Circle.  The  effect  of  the  addition  of  the  term 
mx  to  f(x),  in  the  equation  y  =  f(x),  has  been  shown  in  §38 
to  change  the  shape  of  the  locus  by  lamellar,  or  shearing,  motion 
in  the  Xy-plane.  We  usually  speak  of  this  process  as  "the  shear  of 
the  locus  y  =  fix)  in  the  line  y  —  mx."  When  appUed  to  the  circle 
?/  =  +  V'a^  —  x'^  the  effect  is  to  move  vertically  the  middle  point 
of  each  double  ordinate  of  the  circle  to  a  position  on  the  line 
y  =  mx.  The  result  of  the  shearing  motion  is  shown  in  Fig.  184. 
The  area  hounded  by  the  curve  is  unchanged  by  the  shear. 

The  equation  after  shear  is  • 

y  =  mx  +  ■\/a'^  —  x^.  (1) 


This  is  the  same  form  as  equation  (5)  of  §253,  if  we  put  m  = 
and  replace  y  cos  ahyyu    After  the  substitution,  rotate  the  curve 


sma 
cos  a 


440       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§255 

through  45°,  and  replace  ^i  by  y  cos  a.    The  equation  can  then 
be  written 


a^  (1  +  sin  a) 


y^  (1  +  sin  a) 


(2) 


Therefore  the  curve  of  Fig.  184  is  an  ellipse. 

The  straight  line  y  =  mx  passes  through  the  middle  points  of 
the  parallel  vertical  chords  of  the  eUipse 


y  =  mx  + 


(3) 


The  locus  of  the  middle  points  of  parallel  chords  of  any  curve  is 
called  a  diameter  of  that  curve.  We  have  thus  shown  that  one 
diameter  of  the  eUipse  is  a  straight  line.  Since  the  same  reasoning 
applies  to 

y  =  mx+  {h/aWa^  -  x\  (4) 


X'              / 

4 . 

0          /      A 

w 

Fig.  184. — The  ellipse  looked  upon  as  the  shear  of  the  circle  OA  in  a 

Une  M'OM. 


which  may  be  regarded  as  any  ellipse  in  any  way  oriented  with 
respect  to  the  origin,  the  proof  shows  that  the  mid-points  of  arbi- 
trarily selected  parallel  chords  of  an  eUipse  is  always  a  straight 
line. 

266.  General  Equation  of  the  Second  Degree.    The  general 
equation  of  the  second  degree  in  two  variables  may  be  written  in 
the  standard  form 


oa;2  +  2hxy  +  &2/'  +  "igx  +  2/3/  +  c  =  0. 


(1) 


§256] 


THE  CONIC  SECTIONS 


441 


In  treatises  on  Conic  Sections  it  is  shown  that  the  general 
equation  of  the  second  degree  in  two  variables  represents  a  conic. 
Three  cases  are  distinguished  as  follows: 

The  general  equation  of  the  second  degree  represents 


an '  eUipse  if  h^  —  ab  <  0 
a  parabola  ii  h^  —  ab  =  0 
a  h3rperboIa  if  h^  —  ab  >  0. 


(2) 
(3) 
(4) 


To  render  the  above  classification  true  in  all  cases  we  must  classify 

the  "imaginary  ellipse,"  -^  +r^  =  —  1,  as  an  ellipse,  and  other 

degenerate   cases   must  be   similarly  treated.    The   expression 
h^  —  ab  is  called  the  quadratic  invariant  of  the  equation  (1),  so 


Fig.  185. — Confocal  ellipses  and  hyperbolas.     Note  that  the  curves  of 
,  one  set  cut  the  curves  of  the  other  set  orthogonally. 


called  because  its  value  remains  unchanged  as  the  curve  is  moved 
about  in  the  coordinate  plane.  In  other  words,  as  the  locus  (1) 
is  translated  or  rotated  to  any  new  position  in  the  plane,  and  while 
of  course  the  coefficients  of  x^,  xy,  and  y^  change  to  new  values,  the 
function  of  these  coefiicients,  h'^  —  ab,  does  not  change  in  value, 
but  remains  invariant.  The  above  facts  are  not  proved  in  this 
book. 


442       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§256 

266.  Confocal  Conies.  Fig.  185  shows  a  number  of  ellipises 
and  hyperbolas  possessing  the  same  foci  A  and  B.  This  family 
of  curves  may  be  represented  by  the  single  equation, 

in  which  the  parameter  k  takes  on  any  value  less  than  a^,  and  in 
which  a  >  b.     If  fc  satisfies  the  inequality 

k  <  6^ 

the  curves  are  ellipses.    If  k  satisfies  the  inequalities 

¥  <  k  <  a\ 

the  curves  are  hyperbolas.  The  ellipses  of  Fig.  185  may  be 
regarded  as  representing  the  successive  positions  of  the  wave  front 
of  sound  waves  leaving  the  sounding  body  AB,  or  they  may  be 
regarded  as  the  equipotential  lines  around  the  magnet  AB,  of 
which  the  hyperbolas  represent  the  lines  of  magnetic  force. 

Exercises 

1.  Sketch  the  curve  y  =  2x  +  \/4  —  as". 

2.  Draw  the  curve 

X  =  2  cos  B 

2/  =  2  sin  (9  +7r/6). 

3.  Find  the  axes  of  the  elUpse 

X  =  3  cos  6 
2/  =  3  sin  (9  +  7r/4). 

4.  Draw  the  curve  y  =  x  +  \/6x  —  x*. 
6.  Draw  the  curve  y  =  x  +  y/x'  —  6i. 

6.  Show  that  y  =  x  ±  \/6x  is  a  parabola. 

7.  Sketch  the  curve  y  =  (I/2)x  +  Vl6  -  x^    

8.  Sketch  the  curve  2/  =  5x  sin  60°  +  cos  60°-\/25  -x«. 

9.  Discuss  the  curve 

x'/a'  +  y'/b"  -  2{xy/ah)  sin  a  =  cos^  a. 


§256]  THE  CONIC  SECTIONS  443 

Show  that  the  locus  is  always  tangent  to  the  rectangle  x  =  ±  o, 
y  =  ±  b,  and  that  the  points  of  contact  form  a  parallelogram  of 
constant  perimeter  4:\/a'  +  b^  for  all  values  of  a.  Hint:  Compare 
with  equation  (6),  §253. 

10.  Show  that  x  =  a  cos  {8  —  a),  y  =  b  sin  (9  —  or)  represents  an 
elUpse  for  all  values  of  a.  ■ 

11.  Prove  from  equation  (8),  §253,  that  the  distance  from  the 
end  of  the  minor  to  the  end.  of  the  major  axis  of  the  resulting  ellipse 
remains  the  same  independently  of  the  magnitude  of  a. 

12.  Show  that  the  following  construction  of  the  hyperbola  xy'  =  a' 
is  correct.  On  the  —  X-axis  lay  off  OC  =  a.  Connect  C  with  any 
point  A  on  the  F-axis.  At  C  construct  a  perpendicular  to  AC  cut- 
ting the  F-axis  in  B.  At  B  erect  a  perpendicular  to  BC  cutting  the 
-|-  X-axis  at  D.  Through  A  draw  a  parallel  to  the  X-axis  and  through 
D  draw  a  parallel  to  the  F-axis.  The  two  lines  last  drawn  meet  at  P, 
a  point  on  the  desired  curve. 

13.  Explain  the  following  construction  of  the  cubical  parabola 
a^y  =  x'.  Lay  off  OB  on  the  —  F-axis  equal  to  a.  From  B  draw  a 
line  to  any  point  C  of  the  X-axis.  At  C  erect  a  perpendicular  to  BC 
cutting  the  F-axis  at  D.  At  D  erect  a  perpendicular  to  CD  cutting 
the  X-axis  at  E.  Lay  off  OE  on  the  F-axis.  Then  OE  is  the  ordinate 
of  a  point  of  the  curve  for  which  the  abscissa  is  OC. 

14.  Explain  and  prove  the  following  construction  of  the  semi- 
cubical  parabola,  ay^  =  i'.  Lay  off  on  the  —  X-axis,  OA  =  u.. 
From  A  draw  a  parallel  to  the  line  y  =  mx,  cutting  the  F-axis  in  B. 
Erect  at  B  a  perpendicular  to  AB  cutting  the  X-a,xis  at  C,  and  at  C 
erect  a  perpendicular  to  Ou.  The  point-  of  intersection  with  y  =  mx 
is  a  point  of  the  curve. 

Miscellaneous  Exercises 

1.  Show  that  sec^  a(l  +  sec  2a)  =  2  sec  2a. 
sin  a  +  sin  2a 


2. 

Show  that  ;; — j 1 s —  =  tan  a. 

1  "1-  cos  a  -\-  cos  2a 

3. 

oL        ii.  J.  COS  a  -|-  sin  a      cos  a  —  sin  a 

Show  that -. i — -. —  =  2  tan  2a. 

cos  a  —  sm.  a     cos  a  -|-  sm  a 

4. 

„,        ^,    ^  cos  (a  —  ff)        1  +  tan  a  tan  fi 

bnow  that 7 — ;— ts  —  'i 1 1 ;;' 

cos  (a  -f-  (3)        1  —  tan  a  tan  /3 

.  1  -1-  tan"  1 
Show  that  .                   =  sec  a. 

5. 

1-tan-^ 

444      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§256 

6.  Solve  the  equation  sin*  a  —  2  cos  a  +  i  =0. 

7.  Simplify  the  product 

(x  -2  -  v'3)(a;  -  2  -  iVs){x  -  2  +  V3)ix  -2  +  iVS). 

8.  Express  in  the  form  c  cos  (o  —  6)  the  binomial 

30  cos  o  +  40  sin  a. 

9.  Find   tan  6  by  means  of  the  formula  for  tan  (A  +  B),  if 
8  =tan-i  1/2  +  tan-i  1/3. 

10.  Find  sin  9,  if  9  =  sin-i  1/5  +  sin-i  1/7. 

11.  Find  the  "equation  of  a  circle  whose  center  is  the  origin  and 
which  passes  through  the  point  (14,  17). 

12.  Discuss  the  curve 

X  =  aS 

y  =  a(l  —  cos  0). 

•  13.  Graph  on  polar  paper  p^  =  a'  cos  29. 

14.  A  fixed  point  located  on  one  leg  of  a  carpenter's  "square" 
traces  a  curve  as  the  square  is  moved,  the  two  arms  of  the  square, 
however,  always  passing  through  two  fixed  points  A  and  B.  Find 
the  equation  of  the  curve. 

16.  Find  the  parametric  equations  of  the  oval  traced  by  a  point 
attached  to  the  connecting  rod  of  a  steam  engine. 

16.  Prove  that 

tan  (45°  +  t)  -  tan  (45°  -  r)  =  .  ^^f'^!    ■ 

17.  Find  the  quotient  of  (6  -  2i)  by  (3  +  75i). 

18.  Solye  for  y  by  inspection: 

sin  (90°  +  iy)  cos  (90°  -  iy)  +  cos  (90°  +  ^y)  sin  (90°  -  iy)  =  sin  y. 

19.  Write  the  parametric  equations  for  the  circle,  the  ellipse,  and 
the  hyperbola. 

20.  The  length  of  the  shadow  cast  by  a  tower  varies  inversely  as 
the  tangent  of  the  angle  of  elevation  of  the  sun.  Graph  the  length 
of  the  shadow  for  various  elevations  of  the  sun. 

21.  From  your  knowledge  of  the  equations  of  the  straight  line  and 
circle,  graph  y  =  ax  +  y/a^  —  x'^- 

(See  Shearing  Motion,  §37.) 

22.  In  the  same  manner,  sketch  y  =  a  -{■  x  -\-  y/'a^  —  x^- 

23.  Graph  the  curve  y  =  1/x  +  x'.  Has  this,  curve  a  minimum 
point? 


§256] 


THE  CONIC  SECTIONS 


445 


24.  Find  by  use  of  logarithmio  paper  the  equations  of  the  curves 
of  Fig.  186.  These  curves  give  the  amounts  in  ce^ts  per  kilowatt- 
hour  that  must  be  added  to  price  of  electric  power  to  meet  fixed 
charges  of  certain  given  annual  amounts  for  various  load  factors. 

25.  The  angle  of  elevation  of  a  mountain  top  seen  from  a  certain 
point  is  29°  4'.  The  angle  of  depression  of  the  image  of  the  mountain 
top  seen  in  a  lake  230  feet  below  the  observer  is  31°  20'.  Find  the 
height  and  horizontal  distance  of  the  mountain  top,  and  produce  a 
single  formula  for  the  solution  of  the  problem. 

26.  Find  the  points  of  intersection  of  the  curves 

a;2  -I-  2/^  =  4 
y^  =  4x. 

27.  Solve  llOx-*  +  1  =  2\x-'. 

28.  Solve  3(a;  -  7)(a;  -  l){x  -  2)  =  (»  +  2){x  -  7){.x  +  3). 

29.  Solve  sin  x  cos  x  =  1/4. 

30.  By  means  of  a  progression,  show  how  to  find  the  compound 
interest  on  $1000  for  25  years  at  5  per  cent. 


100 

90 
80 
70 

I 
I   50 

•s 

o 
^  40 


\ 

-r 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

s 

S 

s 

^' 

^•> 

\ 

s 

\ 

s 

s 

^ 

■fe, 

s 

■v 

^ 

^ 

^ 

- 

.^ 

!a 

'■a 

1  ] 

^rn 

*-. 

~ 

^ 

^ 

J 

is. 

fSc 

1 

hi 

Vr, 

as 

~ 

~ 

- 

~ 

"■ 

— 

0.1     0.2     0.3     0.4     05 


0.6     0.7     0.8     0.9       1 
Cents  per  K.W.Hour 


11     1.2     1.3     1.4     15 


Flo.  186.— Annual  fixed  charges  of  $10,  $15,  and  $20  of  a  certain  hydro- 
electric plant,  reduced  to  cents  per  kw-hr  for  various  load  factors. 


446       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§256 
31.  Find  the  approximate  equations  for  the  following  data: 


(a)  Steam  pressure : 
(6)  Gas-engine  mixture : 


V  =  volume,  p  =  pressure. 

V  =  volume,  -p  =  pressure. 


(a) 

(6) 

r 

V 

V 

V 

2 

68.7 

3.54 

141.3 

4 

31.3 

4.13 

115.0 

6 

19.8 

4.73 

95.0 

8 

14.3 

5.35 

81.4 

10 

11.3 

5.94 
6.55 
7.14 

71.2 
63.5 
54.6 

32.  Show  that  p^  =  a^  cos  29  is  the  polar  equation  of  a  lemniscate, 

33.  When  an  electric  current  is  cut  off,  the  rate  of  decrease  in  the 
current  is  proportional  to  the  current.  If  the  current  is  36.7  amperes 
when  cut  off  and  decreases  to  1  ampere  in  one-tenth  of  a  second, 
determine  the  relation  between  the  current  C  and  the  time  t.- 

34.  Write  four  other  equations  for  the  circle  p  =  2-\/3  sin  9  — 
2  cos  e. 

36.  Write  four  other  equations  for  the  sinusoid  j/  =  sin  x  — 
■\/3  cos  X. 

36.  Find  the  angle  that  Zx  +  iy  =  12  makes  with  ix  -  Zy  =  12. 

37.  From  the  equation 

9  =  6  sin  (2«  -  1°) 

determine  the  amplitude,  period,  and  frequency  of  the  S.H.M. 

38.  A  simple  sinusoidal  wave  has  a  height  of  3  feet,  a  length  of  29 
feet,  and  a  velocity  of  7  feet  per  minute.  Another  wave  with  the 
same  height,  length,  and  velocity  lags  15  feet  behind  it.  Give  the 
equation  of  each. 

39.  Simplfy 

(3\/3  -  3i)  2(  -  1  -I-  VZiy      (cos  36°  +  i  sin  36°)  (cos  20°  +  i  sin  20°) . 


(2  -I-  2^31) 


2(cosll°  +isin  11°) 


§256]  THE  CONIC  SECTIONS  447 

40.  Calculate      (1  -  VsifK 

41.  Plot  the  amount  of  tin  required  to  make  a  tomato  can  to  hold 
1  quart  as  a  function  of  the  radius  of  its  base.  Deterinine  approxi- 
mately from  the  graph  the  dimensions  requiring  the  least  tin. 

42.  Find  the  axes  of  the  ellipse  whose  foci  are  (2,  0)  and  (  —  2,  0), 
and  whose  directrices  are  x  =  ±  5. 

43.  Write  the  polar  equation  for  the  ellipse  in  problem  42. 

44.  Find  the  equation  of  the  hyperbola  whose  foci  are  (5,  0)  and 
(—  5,  0),  and  whose  directrices  are  x  =  +  2. 

46.  Write  the  equation  of  the  hyperbola  of  problem  44  in  polar 
coordinates. 

46.  Discuss  the  curve  p(l  +  cos  9)  =  4.  Write  its  equation  in 
rectangular  coordinates. 

47.  Find  the  foci  of  the  hyperbola  2xy  =  a".     Also  its  eccentricity. 

48.  Find  the  equation  of  the  locus  of  a  point  whose  distance  from 
the  point  (3,  4)  is  always  twice  its  distance  from  the  line  3x  +  4y  =  12. 
What  is  the  locus? 

49.  A  point  moves  so  that  the  quotient  of  its  distance  from  two 
fixed  points  is  a  constant.  Find  the  equation  of  the  locus  of  the 
point. 

60.  Evaluate  log  10  -  log2  8  +  logy  492. 

51.  Find  the  maximum  and  minimum  value  of  (3  sin  j:  —  4  cos  x). 
What  values  of  x  give  these  maximum  and  minimum  values? 

62.  Find  the  equation  of  a  circle  passing  through  the  points  (1,  2), 
(-  1,  3),  and  (3,  -  2). 

63.  A  sinusoidal  wave  has  a  wave-length  of  ?r,  a  period  of  tt,  and  an 
amplitude  of  t.     Write  its  equation. 

64.  Compute  the  value  of  each  of  the  following : 

1^;    7  ois  47°  X  6  cis  (-  14°);  (7  +61)";    '^Ti+ST. 

65.  Prove  by  the  addition  formulas  that: 

sin  (90°  -t)  =  COST,  sin  (360°  -  t)  =  -  sinr, 
sin  (90°  +t)  =  COST,  tan  (r  +  270°)  =  -  cotT. 


56.  Solve     x2  +  6x  +  Vx'  -|-  6x  -|-  1  =  1- 

57.  Find  the  product  of  3  -  2i  by  -  2  +  i. 
68.  Find  all  the  values  of 

(cos  e  -I-  1  sin  e)2;  (cos  S  +  i  sin  6)^^;  -^V,  VT. 


448       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§250 

69.  Show  that 

sin  (a  +  6  +  c)  =  sin  o  cos  b  cos  c  +  cos  a  sin  b  cos  c 

+  cos  a  cos  b  sin  c  —  sin  o  sin  6  sin  r 

60.  Draw  upon  squared  paper,  using  2  cm.  =  1,  the  curve  y"  =  x 
By  counting  the  small  squares  of  the  paper  find  the  area  bounded  by 
the  curve  and  the  ordinates  x  =  1/2,  1,  IJ,  2,  2i,  3,  3i,  4,  .  .By 
plotting  these  points  upon  some  form  of  coordinate  paper,  find  the 
functional  relation  existing  between  the  x  coordinate  and  the  area 
imder  the  curve. 

61.  The  latitude  of  two  towns  is  27°  31'.  They  are  7  miles  apart 
measured  on  the  parallel  of  latitude.  Find  their  difference  in 
longitude. 

62.  Solve  3"'"'  =  2'+'.  Be  very  careful  to  take  account  of  all 
questionable  operations.     There  are  two  solutions. 

63.  Find  (two  problems)  the  equation  connecting: 


X 

y 

6.8 

19.0 

14.2 

21.6 

21.8 

23.2 

32.0 

26.3 

46.5      . 

31.5 

65.0 

39.1 

78.0 

47.0 

X 

y 

1.3 

21 

2.0 

25 

2.8 

29 

3.7 

33 

4.3 

35 

5.3 

38 

64.  Find  the  wave  length,  period,  frequency,  ampUtude,  and  velocity 

for  y  =  10  sin  (2x  -  30. 

66.  Prove  that 

csc^  A  „  . 

— 5— j ^  =  sec  2A. 

csc^  A  —  2 

66.  Find  the  equation  of  the  elhpse,  center  at  the  origin,  axes  coin- 
ciding with  coordinate  axes,  passing  through  the  point  ( —  3,  5)  and 
having  eccentricity  3/5. 
,  67.  Prove  (esc  2x)  (1  —  cos  2x)  =  sin  x  sec  x. 

(esc  X)  (1  —  cos  x)  =  ? 

68.  A  S.H.M.  has  amplitude  6,  period  3.  Write  its  equation  if 
time  be  measured  from  the  negative  end  of  the  oscillation.  State  the 
difference  between  a  S.H.M.  and  a  wave. 


§250]  THE  CONIC  SECTIONS  449 

69.  Sketch  on  squared  paiper : 


y  = 

V 

y  = 

2' 

y  =  logz  X 

y  = 

3" 

y  =  logs  X 

y  = 

5^ 

y  =  logs  X 

y  = 

10- 

y  =  logio  X 

70.  Solve  3»  -  2.T  =  1. 

71.  Sketch 

p  =  a,            p  =  sec  6, 

p 

= 

a  sin  e, 

1 

p  =  -,             p  =  a  cos  9, 

p 

= 

—  a  cos  9, 

p  =  (2  —  cos  e), 

p 

= 

2  cos  6/  -  3, 

p  =  —  o  sin  9, 

P  =  a  —  a  cos  8, 

P  =  cos  9  +  sin  6. 

72.  Simplify  the  expression 

sin   {^  -  rj  sec  ^1  +j;j    -  sin  Q  +  r^  sec  (|  -  rj 

73.  Simplify  and  represent  graphically 

y«-)\^'^^)  (1  +»)a  +  2i). 

74.  Find  the  coordinates  of  the  center,  the  eccentricity,  and 
the  lengths  of  the  semi-axes  of:  (o)  rc^  +  Sx  +  j/^  =  7,  (6)  x^  -\-  2x 
+  42/2  -  32/  =  0,     (c)  a;2  -  a;  -  2/2  -  2/  =  0,  (d)  s^  -f-s  +  j,  +  3  =  0. 

75.  Knd  the  amplitude,  period,  frequency  and  epoch  of  the  fol- 
lowing S.H.M. 

2/  =  7  sin  6i. 

2/  =  6  sin  2irt. 

y  =  a  sin  {id  -f-  e) . 

76.  Find  cis=  e.     Show  that 

cos  5x  =  cos^  a;  —  10  cos'  x  sin^  i  +  5  cos  x  sin''  x. 

77.  Find  graphically  (on  form  MZ)  the  fifth  roots  of  2^  cis  35°. 

78.  Complete  the  following  equations : 


sin  (a  +  b)  =  ? 

tan  2x  =  t 

cos  (a  ±  6)  =  ? 

cot  2a;  =  ? 

tan  {a  +  B)='i 

sin^  =  ? 

20 

450      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§256 


sin  2a;  ==  ?  cos  -  =  ? 

2 


cos  2x  =  ">  cot  -  =  ? 

2 


79.  Solve  a'  +  1  =  0. 

80.  y  =  —  St'  +  4i  —  5  and  x  =  5t  are  the  parametric  equations 
of  a  curve.    Discuss  the  curve. 

81.  Show  that  [rfcos  e  +i  sin  S)]  [(r'(cos  B'  +  i  sin  B')]  = 
rr'lcos  (e  +  B')  +i  sin  (e  +  6')]. 

82.  Two  S.H.M.  have  amplitude  6  and  period  two  seconds.  The 
point  executing  the  first  motion  is  one-fourth  of  a  second  in  advance 
of  the  point  executing  the  second  motion.  Write  the  equations  of 
motion. 

83.  Show  that  sin  5x  =  sin*  a;  —  10  sin'  x  cos'  x  -\-  5  sin  x  cos^  x. 


CHAPTER  XV 

APPENDIX 

A  REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA 

300.  Only  the  most  important  topics  are  included  in  this  review 
Prom  five  to  ten  recitations  should  be  given  to  this  work  before  begin- 
ning regular  work  in  Chapter  I. 

With  the  kind  permission  of  Professor  Hart,  a  number  of  the  exer- 
cises have  been  taken  from  the  Second  Course  in  Algebra,  by  Wells  and 
Hart. 

,    301.  Special  Products.    A  few  simple  muItipUcations  may  be  per- 
formed mentally. 

(1)  The  product  of  the  sum  and  difference  of  any  two  numbers: 

(a  -I-  6)(a  -  6)  =  o2  -  62 
From  this  we  have  (3a;  -  2y)i3x  +  2y)  =  9x^  -  ^y^. 

Exercises 
Multiply  mentally  the  following : 

1.  (3a;  -  J/) (3a;  -|-  y).  6.  (29)(31),  or  (30  -  1)(30  +  1)  = 

900  -  1  =  899. 

2.  (2a;  +  7)(2a;  -  7).  7.  (51) (49). 

3.  (5a;  -  y){5x  +  y).  8.  (52)(48),  or  (50  +  2)(50  -  2). 

4.  Ixh/  -  3a){x'y  +  3o).        9.  (103)  (97). 
6.  (o  +  3b)  (o  -  36).  10.  (25)  (35). 

(2)  A  few  products  of  binomials  are:  \ 

(o  +  by  =  a'  +  2ab  +  6'. 

(o  -  by  =  o'  -  2o6-|-  6^ 

(a  +  6)3  =  a>  +  3a'b  +  3ab'  +  b\ 

(a  -  6)»  =  a'  -  3a'b  +  3a¥  -  bK 

la  +  h)*  =  a*  +  4o»6  +  6a'b'  +  4o6»  +  6*. 

(a  -  b)*  =  a*  -  4o»6  +  6a'b'  -  4a6'  +  6". 
Thus  (3  -  o)  3  =  27  -  27o  +  9a»  +  a\ 

and  (a;  -t-  y^Y  =  x*  +  4a;'v«+  Qx^y*  +  ^y^  +  y\ 

451 


452       ELEMENTAE/y  MATHEMATICAL  ANALYSIS       [§301 

Expand  mentally  the  following: 

1.  (2o  -  x)K  4.  {x  -  d)*. 

2.  (a;  +  3yy.  5.  (1  -  a;)'. 

3.  (2x  -  1)'.  •  6.  (2  +  yY. 

7.  (52)2,  or  (50  +  2)',  or  2500  +  200  +  4  =  2704. 

8.  (31)2,  or  (30  +  ly.  9.  (29)',  or  (30  -  1)^. 

(3)  The  square  of  a  polynomial  is  illustrated  hy  the  following: 

(a  +  b  +  c)2  =  a"  +  62  +  c2  +  2ab  +  lac  +  2bc. 

(o  +  6  +  c  +  d)2  =  a2  +  62  +  c2  +  d2  +  2o6  +  2oc  +  2ad  +  26c  + 

26d  +  2cd. 
(3  -  a;  +  !/)2  =  9  +  x2  +  2/2  -  6a;+  62/ -  2xy. 

Expand  mentally  the  following : 

1.  (o  +  6  +  2)2.      ■  •  4.  (2a  -  X  +  3)2. 

2.  (a  +  6  -  2)2.  5.  (x2  -  22/2  +  4)2. 

3.  (a  -  6  -  c)2.  6.  (x  -  2o  -  62/2)'. 

(4)  The  product  of  two  binomials  having  a  common  term: 

(x  +  a){x  +  6)  =  x2  +  (o  +  b)x  +,ab. 
Thus  (x  +  5)(x  -  11)  =  x2  +  (5  -  ll)x  +  5(  -  11), 

=  x2  -  6x  -  55. 
(x  +7)(x  +  2)  =  x2  +9x  +  14. 
(x  -  5)(.x  +  3)  =  x2  -  2x  -  15. 
(x2  -  22/)  (x2  -  32/)  =  x"  -  5x22/  +  6j/2. 

Find  mentally  the  value  of  each  of  the  following : 

1.  (x  +  2)(x  +  3).  6.  (3x  +  22/)(3x  -  7y). 

2.  (x  -  2)(x  +  3).  7.  (x2  -  3)(x2  -  4). 

3.  (x  -  2)(x  -  3).  8.  (3x1/  -  z)(3x2/+  7z). 

4.  (x  +  2)(x  -  3).  9.  (x22/2  -  3)(x22/2  -  10). 

5.  (x2  +  52/) (x2  -  52/).  10.  (x  -  2y){2x  -  2y). 

(5)  rfte  product  of  two  general  binomials: 

{ax  +  6)  (ex  +  d)  =  ocx2  +  (6c  +  ad)x  +  bd. 
Thus 

(3o  -  4b) (2a  +  76)  =  (3a) (2a)  +  (-  8  +  21)ab  +  (-  4  b)  (7b) 
=  6a2  +  13ab  -  28b2. 

Find  mentally  the  following  products: 

1.  (5x  -  22/)2.  4.  (2m  +  3)(m  +  4). 

2.  (a  +  llb)(a  +  36).  5.  (2/2  +  4z)(2/2  +  4z). 

3.  (a  -  2u)(a  +  12»).  6.  (3x2/  -  7)2. 


§302]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     453 

7.  (Sw^w  -  4:)(.3uh;  +  4).  29.  (2  -  3s«)(5  +  2st). 

8.  {2x  -  5)(a;  +  4)i  30.  (a^b  +  6c)  (0^6  -  13c). 

9.  (2r2  -  7)(3r2  +  5).  31.  [Ip  +  5)(lp  -  4). 

10.  (p2  -  Sq){,p^  +  7q).  32.  {a'  +  7)(o'  -  11). 

11.  (a  +  l)(o  -  i).  33.  (So  +  5)  (7a  -  8). 

12.  iix  +  5y){ix  -  By).  34.  (1  +  8n)(l  -  9n). 

13.  (u  -  |)(w  -  I).  35.  (2a  -  6") (2a  +  3b*). 

14.  (2a;  +  3)(Js  +  1).  36.  (12a;  -  i){9x  -  J). 
16.  (3x2  4.  46c) (3x'  -  46c).  37.  (20  -  16z)(3  +  2z). 

16.  iy  -8)(.y  +  5).  38.  (r^  +  16s)  (r^  -  s). 

17.  (X  -  i){x  -  f).  39.  (a  -  6x2) (a  4.  ^2^ 

18.  (1  -  6s) (3  +  2s).  40.  (4r  +  uv)(ir  -  5uv). 

19.  (2<  -  '7w^){3t  -  4u)2).  41.  (6x2  _  1)2, 

20.  (|u  -  i)(f«  +  J).  42.  (1  +  23n)(5  -  n). 

21.  (3r  -  7<)(5r  +  2t).  43.  (x*  -  2/*)(x«  +  y*). 

22.  (11x2  _  I)(i2x2  +  1).  44.  (5a2  -  4b)(6a2  -  56). 

23.  (z2  -  6) (02  +  12).  46.  (x2j/  +  yH){xh/  -  y^x). 

24.  (x  +  32/2) (x  -  22/2).  46.  (fa  +  10) (2a  +  1). 
26.  (6m»  -  6s2)(5m'  +  s2)  47.  (9r  +  2s)  (3r  -  4s). 

26.  (5x  +  |)(5x  -i).  ■  48.  (12x2  4.  5)  (43,2  _  3). 

27.  (3x  +  7)(x  -  5).  49.  (a26*  +  4x2)2. 

28.  (4o  -  363)2.  60.  (a^  -  5«)(a«  +  6«). 
61.  (a  +  6)(a  -  6)(a2  +  62)(a*  +6<). 

302.  Symbols  of  Aggregation.  If  a  sign  of  aggregation  is  preceded 
by  the  negative  sign,  change  all  signs  within  when  the  sign  of  aggre- 
gation is  removed.  If  the  sign  of  aggregation  is  preceded  by  the  posi- 
tive sign,  all  signs  within  remain  the  same  when  the  sign  of  aggre- 
gation is  removed. 

5x2  _  [syi  4.  {2x2  _  (2,2  4.  3^2)  4-  5j/2}  _  ^2; 

=   5X2    _   [■^yl   4.    {2x2    -   y2    -   3x2   4.   52^2}     _   3;2] 
=  5X2    _   [3y2  4.    {42,2    _  3.2}     -  x2] 
=   6x2    _    [3y2   _|_  4j,2    _   -j2    -   x2] 
=    5X2    _    [7j,2    _   2x21 
=    5x2    _   7y2   4.  2x2 

=     7X2     _     -Jyl 

Exercises 

Simplify  the  following  by  removing  the  signs  of  aggregation : 

1.  ab  -  46^  -  (2a2  -  62)  _  {  _  502  4.  206  -  862). 

2.  X  -  { 2/  -I-  z  -  [x  -  (  -  X  -  2/)  -f  z])  +  [z  -  (2x  -  2/)]. 


454      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§303 

3.  o-{  -o  —  [-o-(-o-  1)]). 

4.  Syz  -  [2yz  +  (9z  -  2yz)]. 

6.    -  {  -1  -[-1  -(-1)]1. 

6.  5x'  -  [Sy'  +  {2x^  -  {y'  +  Zx'')  +  5y^}  -  x']. 

7.  ab  -  [46^'  -  (2o»  -  b^)  -  [  -  5o'  +  2ab  -  3b']]. 

8.  33/'  -  I2y'  +  (9z  -  2yx)]. 

303.  Factoring.  Since  (a  +  b)'  =  a'  ±  2ab  +  b',  any  expression 
of  the  form  of  the  right-hand  side  can  be  factored  by  inspection. 
Thus, 

x'  -  6xy  +  9y'  =  {x  -  3y)' 
and 

4  +  4(o  +  6)  +  (a  +  6)2  =  (2  +  o  +  6)' 

Exercises  1 

Factor  the  followlag  by  inspection: 

1.  Qx^  -  ZQxy  +  25yK 

2.  4  +  16«  +  16«». 

3.  a;*j/*  +  10a;'j/2z2  +  25z*. 

4.  9  +  6(x»  +  j/»)  +  (I'  +  !/>)».      . 
6.  a*  +  4o26«  +  4b<. 

Since  (o  +  6)  (o  —  6)  =  a'  —  6',  any  expression  of  the  form  of  the 
right-hand  side  can  be  factored  by  inspection.     Thus, 

4o2  -  9b'  =  (2a  -|-  36)  (2o  -  36). 

Exercises  2     j 
Factor  the  following  by  inspection: 

1.  x'j/'  -  «'•  4.  25  -  3a;'. 

2.  (o  -I-  5)'  -  c'.  6.  81  -  625x*. 

3.  c^  -  {a-\-  6)'. 

Since  (a  4-  6)(o  +  c)  =  o'  +  (6  +  c)a  +  6c,  any  expression  of  the 
form  of  the  right-hand  side  can  be  factored  by  inspection.    Thus, 

a;'  -5a;  -  14  =  (a;  -  7)(a;  +2) 

Exercises  3 
Factor  the  following  by  inspection: 

1.  a;'  +  7s  +  10.  4.  9i'  -  18s  -  27. 

2.  a'  +  4aj/  -  21?/'.  5.  25  +  30o  -  27a». 

3.  4a;'  -  18iy  +  18i/'. 


§303]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     455 

Since  (a  +  &)(«'  —  ah  +  6*)  =  a'  +  b',  any  expression  of  the  form 
of  the  right-hand  side  can  be  factored  by  inspection.    Thus, 

27  +  125a;»  =  (3  +  5x)(,Q  -  ISa;  +  25a;2). 


Exercises  4 

Factor  the  following  by  inspection: 

1.  x'y'  +  1.  4.  125  +  x'yK 

2.  x'  +  y".  5.  x'  +  8yK 

3.  8  +  27a;'.  i 

Since  (o  —  6)(a^  +  ab  +  6')  =  o'  —  6',  any  expression  of  the  form 
of  the  right-hand  side  can  be  factored  by  inspection.     Thus, 

27  -  125a:'  =  (3  -  5x)(9  +  15a;  +  25ai2). 


Exercises  5 

Factor  the  following  by  inspection: 

1.  x>y'  -  1.  4.  125  -  x^yK 

2.  x^  —  y^,  or  (a;*  +  y^)ix'  —  y^).    5,  z^  —  8yK 

3.  8  -  9a;'.  6.  27  -  8a'; 

The  following  may  be  factored  by  grouping  the  terms.     Thus, 

a'm  +  o're  —  m  —  n  =  a'(m  +  n)  —  (m  +  ra) 
=  (o'  —  l)(m  +  n) 
=  (a  -  l)(a2  +a  +  l)(m  +  n). 


Exercises  6 

Factor  the  following: 

1.  ax  —  ay  +  bx  —  by.  4.  x^  —  xy*  —  x^y  +  y^. 

2.  a;'  +  3a2  +  3a;  -  1.  5.  a;*  -  x^y  -  xy^  +  y\ 

3.  ax^  -  2axy  +  ay'  +  bx'  -  2bxy  +  by', 

A  trinomial  of  the  form  px'  +  gx  +  ?■,  if  the  product  of  two  bino- 
mials, may  be  factored  as  outlined  below. 
In  the  product 

ax  +  b 
ex  +  d 
OCX'  +  {be  +  ad)x  +  bd 


456       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§303 

the  terms  acx''  and  hd  are  called  end  prodiicts  and  bcx  and  adx  are  called 
cross  proditcts.  This  most  important  case  of  factoring  is  best  learned 
from  the  consideration  of  actual  examples. 

Factor  21x'i  +  5a;  -  4. 

Prom  the  term  21a;*,  consider  as  possible  first  terms  7s  and  3a;,  thus 
(7a;  )(3a;  ).  For  factors  of  (—  4),  try  2  and  2,  with  unhke  signs, 
and  signs  so  arranged  that  the  cross  product  with  larger  absolute 
value  shall  be  positive;  thus  (7a;  —  2)(3i  +  2).  This  gives  middle 
term  Sx;  incorrect.  For  (—4)  try  4  and  1,  with  signs  selected  as  be- 
fore; thus,  (7x  —  l)(3a;  +  4).  Middle  term  25a;;  incorrect.  Try 
(7a;  +  4) (3a;  —  1).     Middle  term  5x;  correct. 

Factor  2ix'  -  17xy  +  ZyK 

Try  (6a;  —  32/)(4x  —  y).  Incorrect,  since  first  ()  contains  factors 
and  given  expression  does  not.  Try  (fix  —  y){4:x  —  3y).  Middle 
term  -  22;  incorrect.  Try  (8a;  -  3y)(3x  -  y).  Middle  term  -  17; 
correct. 


Exercises 

1  7 

Factor  the  following: 

1.  6x'  -  7a;  +  2. 

8. 

35u2  +  UV  -  6t)2. 

2.  3x2  -i-  8x  +  4. 

9. 

9*2  -  14*  -  8. 

3.  6x2  -  a;  -  2. 

10. 

121^^  -  35x2/  -  32/2. 

4.  9a2  +  15o  +  4. 

11. 

6  -  i  -  15*2. 

5.  66"  -  76  -10. 

12. 

5  +  9s  -  18s2. 

6.  14x2  +  13^  _  i2y\ 

13. 

24m2  -  17mn  +  3n2. 

7.  8z2  -  2yz-  2lyK 

14. 

28y"  -  yz  -  2zK 

An  expression  of  the  from  o*  +  0252  -\-  b*  may  be  put  in  the  form  of 
the  difference  of  two  squares  by  adding  and  subtracting  a^b'.     Thus, 

a*  +  a'b'  +  b<  =  a*  +  2a2b2  +  (,2  _  a^b^ 
=  (a2  +  62)2  _  a'b" 
=  (a2  +  ab  +  62) (a2  -  ab  +  b'). 

Exercises  8 

Factor  the  following: 

1.  X*  +  x'y'  +  yK  5.  16x*  +  36x'y^  +  81yK 

2.  X*  +  4x2  4.  16.  6.  a*  +  a*V  +  b*. 

3.  2/*  +  iy'z^  +  I62  .  7.  aV  +  a^x'y'  +  y\ 

4.  16  +  4«2  +  u\  8,  625x«  +  100x2z<  +  162».     . 


§304]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     457 

304.  To  factor  a  polynomial  completely,  first  remove  any  monomial 
factor  present;  then  factor  the  resulting  expression  by  any  of  the  type 
forms  which  apply,  until  prime  factors  have  been  obtained  throughout. 
'Thus, 

(a)  5a«  -  5&«  =  5{a^  -  6«)  =  b{cfi  -  V){a^  +  6=) 

=  5{a  -  b)(a2  +  a6  +  V){a  +  h){a^  -  ah  -^  h") 

(b)  42aa;2  +  lOox  -  8a  =  2a{2\x^  +  5x  -  4) 

=  2a(Jx  +4)  (3a;  -  1) 

(c)  2Cmnu^  -  IWmnu  +  X2imn  =  5mn{^'  -  20m  +  25) 

=  bmnhu  -  5Y. 

Exercises 

Factor  the  following  expressions:  , 

1.  xV"°  -  A"*-  22.  a;2  +  Qx  -  27. 

2.  9x»  -  43/6.  23.  c'  -64«3. 

3.  ,25a;«  -  1.  24.  Sx'  -  1. 

4.  81  -  ^K  26.  1  -  13<  -  68«2. 
6.  1  -  6ia''b*c\                            26.  a;<  -  Cx^b  -  SSb". 

6.  a;'  —  y^.  27.  au"  —  4aM!;  —  i5av^. 

7.  225    -  aS.  28.  28a2  -  a  -  2. 

8.  121x2  -  1442/2.  29.  Ss^  -  17si  +  24{2. 

9.  49ot«  -  SQx'y^zK  30.  15r=  -  r  -  6. 

10.  169  -:^  a'lx^.  31.  iy^  -  3y  -  7. 

11.  4x2  _  20x  +  25.  32.  641*6  _  27x3. 

12.  9o2  +  6ob  +  b2.  33.  6ar  -  3as  +  4a«. 

13.  a'b^  -  nabc  -  QOcK  34.  a^  +2a  -  35. 

14.  r*  -  llr'  +  30.  35.  9x2  ^  i2xy  -  32^". 
16.  16b2  +  30b  +  9.  36.  o"  +  lOab  +  25b2. 

16.  Slu"  +  180ua  +  lOOs;'  37.  625x22/2  -  ^. 

17.  36a2  -  l32o  +  121.  38.  3cdy'  -  9cdy  -  30cd. 

18.  x'y*  -  Axy^  +  4.  39.  4ox2  -  25ay*. 

19.  o2b2  -  2ab  -  35.  40.  3y^  +24. 

,20.  u'  +~u3  _  110.  41.  4x2  _  27x  +  45. 

21.  a*b2  -  14o2b  +  49.  42.  6x2  +  7^  _  3, 


458       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§305 

43.  -jV'  -  1.  58.  2am«  -  .50a. 

44.  10x>y  -  5x^y^  -  5xy\  69.  72  +  7a;  -  49a;». 

45.  »i»n»  +  7mn  -  30.  60.  31a;'  +  23xy  -  8yK 

46.  x^  -  Zxy  -  70yK  61.  24o»  +  26a  -  5. 

47.  mx"  +  7mx  -  44r«.  62.  1  -  3xy  -  IQSx'y^ 

48.  x'  -  3a;»  -  108x.  63.  x^  -  Umx  +  AOm^. 

49.  x>  -  yK  64.  26  +  10a5  -  28o%. 

60.  a;*  -  hx-^  -  'iAy\  65.  c»  +  27(f». 

61.  8n«  +  18n  -  6.  66.  Zx^y  -  27xy\ 

62.  3i*  -  12.  67.  -^^^  -  4^^*- 

63.  Stw"  -  42ot«  +  49<».  68.  49ji<2/  -  196nV- 

64.  lOa;'  -  39a;  +  14.  69.  a;«  -  16a;  +  48. 

65.  12x«  +  11a;  +  2.  70.  a;'  +  23a;  -  50. 

66.  363;"  +  12i  -  35.  71.  a<w«  +  31a!^2  +  30. 

67.  x'  -  SyK  72.  9a;'  +  Z.7xy  +  iy\ 

306.  General  Distributive  Law  in  Multiplication.  From  the  mean- 
ing of  a  product,  we  may  write 

(a  +  6+c+.    .    .){x  +  y  +  z  + .    .    .)=ox  +  6a;  +  ca;-|-.  .  . 

+  ay  +  hy  +  cy  +.  .  . 

+  az  +  bz  +  cz  +.  .  ., 
etc. 

Stating  this  in  words :  The  product  of  one  polynomial  by  another  is  the 
sum  of  all  the  terms  found  by  multiplying  each  term  of  one  polynomial 
by  each  term  of  the  other  polynomial. 

To  multiply  several  polynomials  together,  we  continue  the  above 
process.  In  words  we  may  state  the  generalized  distributive  law  of 
the  product  of  any  number  of  polynomials  as  follows: 

The  product  of  k  polynomials  is  the  aggregate  of  all  of  the  possible 
partial  products  which  can  be  made  by  multiplying  together  k  terms,  of 
which  one  and  only  one  must  be  taken  from  each  polynomial. 
Thus, 

{a  +  b+c-V.    .    .){x+y  +  z+.    .    .)(«+«  + 10 +.    .    .) 
=  axu  +  axv  +  .    .    .  +  ayu  +  ayv  +  .    .    .  +  azu  +  azv  +   .    .    . 
+  bxu  +  bxv  +  .    .    .   +  byu  -H  byv  -|-  .    .    .  +  bzu  +  bzv  +  .    .    . 
+  cxu  +  cxv  +  .    .    . 
4-  .    .    .,etc. 

If  the  number  of  terms  in  the  different  polynomials  be  n,  r,  s,  t.   .    . 
respectively,  the  total  number  of  terms  in  the  product  will  be  nrst  ,    ,   . 
The  student  may  prove  this. 


§306]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     459 

306.  The  Fundamental  Theorem  in  the  Factoring  of  z"  ±  a".    The 

expression  (s"  —  a")  is  always  divisible  by  (x  —  a),  when  n  is  a  posi- 


Write  a"— a»  =  s"  —  ox"~i  +  oa;""^  —  a" 

=  a;"~'(a;  —  a)  +  oCa;"""-  —  o"~^) 

Nowi/(a;'~'  —  o'~i)  is  divisible  by  {x  —  o),  then'plainly  s'~'(a;  —  a) 
+  a(a;'~'  —  a'~i)  is  also  divisible  by  {x  —  a).  But  this  last  expression 
equals  (x*  —  o*),  as  we  have  shown.  Therefore,  if  (x  —  a)  exactly 
divides  (a;*~i  —  o*~'),  it  will  also  exactly  divide  (a*  —  o*). 

That  is,  if  the  law  is  true  for  any  positive  integral  value  of  k,  it 
is  true  for  k  one  greater.  But  by  actual  division  the  law  is  true  when 
k  is  3,  (x'  —  o'  =  (x'  +  ax  +  a'){x~a)  therefore  it  is  true  when 
k  is  4.  Being  true  when  k  is  4,  it  is  true  when  k  is  5,  and  so  on  up 
to  fc  =  n,  any  positive  integer. 

We  see  that  (x  —  o)  is  one  factor  of  (x"  —  o").  The  other  factor  of 
(x"—  o»)  is  found  by  actually  dividing  (x"  —  a")  by  (x  —a).  Thus 
(x»  -  a")  =  (x  -  o)(x"-i  +  ax"-'  +  a'x"-^  +  .    .    .  +  o»-=x  +  o"-') 

The  student  may  show  that  (x  +  a)  divides  x"  +  o"  if  ra  be  odd,  and 
divides  x"  —  a"  if  n  be  even. 

Exercises 

Factor  the  following: 

1.  x'  +  yK  7.  m*  -  243. 

2.  x5+32.  8.  32o»  +2436^ 

3.  x«  -  81.  9.  64  -  x«. 

4.  x»  +  1.  10.  x'y'  -z». 
6.  X*  -  162/*.  11-  a:'  -  2/". 

6.  x'j/s  +  1.  12.  27x5  -  8y\ 

Miscellaneous  Exercises  in  Factoring 

Factor  the  following: 

1.  {a+by  -  c\  10.  (1  +«!=)«-  iuK 

2.  (to  -  n)2  -  x\  11.  9(to  -  n)"  -  12(m  -  n)  +  4. 

3.  (x  -  2/)2  -  z2.  12.  (xii  -  4)2  -  (x  +  2)2. 

,  4.  x2  -  (v  -  z)2.  13.  (x2  +  3x)2  +  4(x2  +  2x)  +  4. 

6.  (7x  -  2yY  -  y'.  14.  (Ox^  +  4)2  -  144x2. 

6.  (a  +  6)2  +  23(a  +  b)  +  60.    15.  (x  +  y^  +  7(x  +  2/)  -  144. 

7.  (x  +  y)'  +  2(x  +y)  -  63.     16.  (a2  +  o  +  9)'  -  9. 

8.  (x  -  yy  -  (x  +  2/)2.  17.  (x  +  2/)»  -  z^ 

9.  (x'  -  22/)2  +  2(x2  -  22/)  + 1.  18.  (x  +  y)'  +  z». 


460       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§307 

19.  x^  -  {y  +  z)K  33.  9a*  -  ix^  +.j/»-  6x'y  -  20x«  - 

20.  x^  -  \y  -  z)'.  IhzK 

21.  aj=  +  {y  -  2)3.  34.  4i/*  -  322/2  +  1. 

22.  (to  +  «)s  +  8«'.  36.  94"  -  31««x«  +  25a;*. 

23.  (re  +  y)'  +  (a;  -  y)'.  36.  25o*  +  340^62  +  496*. 

24.  27a»  -  (o  -  b)».  37.  2a(a;  +  i/)  -  3(a;  +  y). 
26.  o'  -  2a6  +  6*  -  c^  38.  a(s  -  j/)  -  6(a;  -  y). 

26.  a;«  +  2xy  +  2/^  -  z^.  39.  ab  +  on  +  6to  +  mn. 

27.  a^  -  a;2  -  2a;2/  -  j/^.  40.  2  +  3a;  -  Sa;^  -  12a;3. 

28.  a;2  -  2/2  -  z2  +  2j/z.  41.  56  -  32a  +  21a2  -  120^. 

29.  b"  -  4  +  2a6  +  02.  42.  4a'  +  o^b^  -  4b=  -  16ab. 

30.  2mn  -  m^  +  1  -  m^.  43.  si  -  sr  -  r"  +  r«. 

31.  9a2  -  24ab  +  16b«  -  '^e.        44.  a;^  +  a;2  +  a;  +  1. 

32.  4a2  -  6b  -  9  —  b^. 

307.  Fractions.  Multiplying  or  dividing  both  numerator  and 
denominator  of  a  fraction  by  the  same  number,  excepting  zero,  does 
not  change  the  value  of  the  fraction.  To  reduce  a  fraction  to  its  low- 
est terms  factor  both  numerator  and  denominator  and  then  divide  out 
the  common  factors  if  there  are  any.     Thus, 

ai*+  aa;2j/'  +  a2/*  _  a{x'^  +  xy  +  y')(3;'  —  xy  +  j/')  _  x'  —  xy  +  y' 
aV  —  a'y^  a'(x  —  y)(x'  +  xy  +  y')        "       a(x  —  y) 

Exercises 

Reduce  the  following  to  lower  terms: 

ax  +  ay  —  X  —  y 


1. 

ax^ 

-  ay^ 

aV 

-a'y^ 

2. 

a;»  + 

X  -6 

x^ 

-  4     ■ 

3. 

X*  - 

y 

4. 
6. 
6. 


x^  +  y^ 
27  -X' 
12  -  7x  +a;2- 
as  -  b* 


a"  -  V' 


Miscellaneous  Exercises  in  Fractions 


1. 


SimpUf y  the  following : 
x^  -  36  7n2 


4n2         n^  +n  -  42 
„    x'  -  2x  -  35      4x2  _  Qx 
'      2x8  _33;2      ■  7(^  _7)- 

„    (5a  +2)  (a  -2) 
•     2a2  +  a  -  10    ■ 


§308]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     461 
4o2  +  8a  +  3    6a2  -  9o 


i. 


6. 


2o2  -  5a  +  3     4a2  -  1  ■ 

16a;  -   4  _  20a;  +  5      a;^  +  2a:  +  1 

5a;  -  5  '  6a:  +  6  "  I6x^  -  1  ' 
a:»  +  8y^  _  X  -2y  _  a;"  +  23;;/  +  4^^ 
a;'  -  8^'  '  a;  4-  22/  '  x^!  -  2a;2/  +  iy"' 

2n'  -n  -  3       n^  +  4»i  +  4       ra^  - 


10. 


n*  -  8n2  +  16  n^  +  n  2n'  -  3n 

x^  —  xy  —  2y^  ^  x  —  2y  ' 

x'  —  9xy^        '    X  —  3y 
2a;g  -xy  -  3y'    ___      3x^  +  xy  -  2y' 

9x'  -  25y'         '    9x'  -  30a;^  +  25^8' 
2a'  -  Sab  -  36=    .   r2a'  -  7ab  -  46=  .  a'  -  4^ab  +  4an 


■  L  a'  - 


a2  -  o6  -  262      ■   la'  -  3ab  -  46'    '    o^  -  ab  -  66^ 

3     \ 


/^+2      _x_\  /^^ 

V     X      ^  X  -  3/  \a:  -  2      x  +  3/ 


1 

X 

308.  Simple  Equations.  Adding  the  same  number  to  both  mem- 
bers of  an  equation  does  not  change  the  equation.  It  follows  that  a 
term  may  be  transposed  from  one  member  of  an  equation  to  the 
other  member  provided  its  sign  is  changed.     Thus  from 

3x  -  2  =  3  +  2x. 
3x  -  2x  =  3  +  2,  or  X  =  5. 

Exercises 

Solve  the  following  equations  for  x: 
1     ^  ~  ^    4.  2x  -  1 


X  -3 

,    2x  -  1 
"•"  4x  -  3 

2x  +  1 

6x  +7 

3 

3 

X  +  2 

X  -2 

2x  +  3 

2x  +  ■. 

o  a;  -p  ^ 

„    X  -  2         2x  +  3 

^-  ~3  r~  -  "■ 

4.  (x  +  4)(x  -  2)  =  (x  +  3)(3x  +  4)  -  (2x  +  l)(x  -  6). 
6.  (,v'  -2x  +  ly  =  (x  -  iy{x  -  3)". 


462      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§309 

309.  Quadratic  equations  are  usually  solved  (a)  by  factoring,  (6) 
by  completing  the  square,  or  (c)  by  use  of  a  formula. 

(a)  To  solve  by  factoring,  transpose  all  terms  to  the  left  member  of 
the  equation  and  completely  factor.  The  solution  of  the  equation  is 
then  deduced  from  the  fact  that  if  the  value  of  a  product  is  zero,  then 
one  of  the  factors  must  equal  zero.    Thus 

(1)  Solve  the  equation 

a;2  +  54  =  15a; 

Transposing  x^  —  15a;  +  54  =  0 

Factoring  (a;  -  9)  (a:  -  6)  =0 

a!-9=0ifa;=9 

X  -  6  =  Qiix  =  6 

Hence  the  roots  of  the  equation  are  9  and  6 

Check:    Does  (9)'  +  54  =  15  X  9? 
Does  (6)«  +  54  =  15  X  6? 

(2)  Solve  the  equation 

12a;2  +  x  =  & 

Transposing  12a;''  +  a  —  6  =0 

Factoring  (3a;  -  2)  (4a;  +  3  =  0 

3a;-2=0ifa;  =  | 
4a;  +'3  =  0  if  a;  =  -f 

Hence  the  roots  of  the  equation  are  f  and  —  f. 

Check:  Does  12(f)2  +  f  =  6? 
Doesl2(-f)»-f=6? 

(b)  To  solve  by  completing  the  square,  use  the  properties  of 
(a;  ±  a)*  =  a;'  ±  2oa;  +  o*,  as  follows: 


REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     463 

(3)  Solve  x^  -  V2.X  =  13. 

Add  the  square  of  1/2  of  12  to  each  side 

a;2  -  12a;  +  36  =  49 

Take  the  square  root  of  each  member 

a -6  =  ±7 

Hence 

a;  =  6  +  7  =  13 
a;  =  6  -7  =  -1 

Check:    Does  (13) ^  -  12  X  13  =  137 

Does  (-1)2  -  12  X  (-1)  =  137 

Since  in  general  (a;  —  a){x  —  V)  =  a;^  —  (a  +  b)x  +  o6,  we  can  check 
thus: 

Does  13  +  (-  1)  =  -  (-  12)7 

Doesl3(-l)  =  -137 

(4)  Solve  x^  -  20a;  +  97  =  0. 

Transpose  97  and  add  the  square  of  1/2  of  20  to  each  side: 
a;'  -  20a;  +  100  =  -  97  +  100  =  3 
Take  the  square  root  of  each  number: 

a;  -  10  =  +  -\/3 
Hence 

si  =  10  +  -\/3[ 
a;2  =  10  -  ^3 

Check:     Does  xi  +  Xj  =  —  (—  20)7 
Does  xiXz  =97? 

(c)  To  solve  by  use  of  a  formula,  first  solve 

0x2  +  6x  +  c  =  0  (1) 

The  roots  are 

-  h±y/V  -  4ac 


2a 


(2) 


For  a  particular  example,  substitute  the  appropriate  values  of  a,  6, 
and  c.     Thus: 

(5)  Solve  2x2  -  3x  -  5  =  q. 

Comparing  the  equation  term  by  term  with  (1)  we  have 

o  =  2,  6=  -3,  c=  -5 


464       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§309 

Substitute  these  values  in  the  formula  (2) 

^  _  -(-3)  + V(-3)'-4(2)(-5) 
2(2) 
3  +  7 


Therefore 


xi  —  5/2,  X2  =  —  1 


Check:     Does  Xi  +  x^  =  —  b/a  =  3/2? 
Does  X1X2  =  c/a  =  —  5/2? 

Exercises 

Solve  the  following  quadratics  in  any  manner: 

1.  s=  +  5x  +  6  =  0.  29.  3i2  _  I2ax  =  63a'. 

2.  a;2  +  4a;  =  96.  30.  ix'  -  I2ax  =  16a^ 

3.  x'  =  110  +  X.  31.  x^  -  X  =6. 

4.  x'  +  5x  =  0.  32.  x'  +7x  =  -  12. 
6.  6x2  +  7a;  +  2  =  q.  33.  x'  -  5x  =  14. 

6.  8x2  -  lOx  +  3  =  0.  34.  x^  +  x  =  12. 

7.  x'  +  mx  -  2m,'  =  0.  36.  x"  -  x  =  12. 

8.  3«2  -  «  -  4  =  0.  36.  x'  =  Qx  -  5. 

9.  107-2  +  7r  =  12.  37.  x'  =  -  4x  +  21. 

10.  x'  +  2ax  =  b.  38.  x^  =  -  4x  +  5. 

11.  x2  +  4x  =  5.  39.  x2  +  5x  +  6  =  0. 

12.  x2  +  6x  =  16.  40.  x'  +  llx  =  -  30. 

13.  2x2  -  20x  =  48.  41.  x2  -  7x  +  12  =  0. 

14.  x2  +  3x  =18.  42.  x2  -  13x  =  30. 

15.  x2  +  Sx  =  36.  43.  3x2  +  4^  =  7. 

16.  3x2  4.  6x  =  9.  44.  3x2  +  61  =  24. 

17.  4x2  _  4a;  =  8.  45.  4-^2  -  5x  =  26. 

18.  x2  -  7x  =  -  6.  46.  5x2  _  7^;  =  24. 

19.  x2  -  ax  =  6a2.  47.  2x2  -  35  =  3x. 

20.  x2  -  2ax  =  3a2.  48.  3x2  _  50  =  5x. 

21.  x2  -  X  =  2.  49.  3x2  _  24  =  6x. 

22.  x2  +  X  =  a2  +  o.  50.  2x2  -  Sx  =  104. 

23.  x2  -  lOx  =  -  9.  51.  2x2  ^  iqx  =  300. 

24.  2x2  _  15a;  =  50.  52.  3x2  _  iqx  =  200. 
26.  x2  +  8x  =  -15.  53.  4x2  -  7x  +  ^  =  q. 

26.  3x2  +  12x  =  36.  54.  |x2  _  f x  =  -  ^^. 

27.  2x2  +  lox  =  100.  55.  9x2  +  6x  -  43  =  0. 

28.  »2  _  5a;  =  _  4,  66.  18x2  -  3x  -  66  =  0. 


§309]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     465 


67. 

|x2  -  3i  +  il  =  0. 

59.  2x2  _  22x  =  -  60. 

68. 

X*       3a; 
4-1+2=0. 

60.  3x»  +  7x  -  370  =  0. 

61.  5x'  -ix-^Ts  =  0. 

*                    -      x^ 

x 

1 

62.3 

-2+i-  =  o- 

63. 

x'  +  2x  +  I  =  Qx  +  6. 

69.  s2  =  5s  +  6. 

64. 

s^  -  49  ^  10(a;  -  7). 

70.  r'  +  3r  =  4.         --^ 

66. 

2x'  +  60a;  =  -  400. 

71.  2s2  +  4as  -  c  =  0. 

66. 

qs  +  7a  +  7  =  0. 

72.  x"  +  6ox  -  5  =  0. 

67. 

z'  =  3z  +  2. 

73.  x2  -  lOax  =  -  9oa. 

68. 

r  =  r2  -  3. 

74.  ca;2  +  2ix  +  e  =  0. 

75.  2x- 

'  +  6x 

-  n  =  0. 

76. 

y'  +  h  =  l 

83.  4x2  -  3x  =  3 

77. 

x^  =  6  +  4a;. 

84.  W  +  4<  =  6. 

78. 

M«  -  fu  -  1  =  0. 

85.  5(x2  -  25)  =  X  -  5. 

79. 

t'+it=  I 

86.  9i42  +  18m  +  8  =  0. 

80. 

r»  -  f  =  ir. 

87.  x*  +  px  +  3  =  0. 

81. 

s^-is=  ^. 

88.  X*  -  8x2  +  15  =  0. 

82. 

3r'  -2r  =  40. 

89.  M*  -  29m2  +  100  =  0. 

2                  8 

5               8 

90. 

X'  ~           3x' 

93.5_,  +  8f,-3. 

91. 

^y  +  i  =  ly 

n-3       n+i_ 
n—  2           n          2 

92. 

*  +  4a;        ix' 

95.3x  +  ^^  +  ^^=2  + 

X 

-      24 

24 

96.- 

X  — 

2  +  1=0. 

97.  x« 

-  35x= 

+  216  =  0. 

98. 

.■  +  f  — 

99.   (a 

,    1\2       16/      ,    1\     ,    ^ 
'+x)   -   3-(^+xj   +7  = 

=  a  -\ 

100. 

u 

a 

101 

2a;  -  7         lOx  -  3 

33 


i 


x2  -  4         5x(x  +  2)' 
Hint:  Clear  of  fractions  by  multiplying  both  members  of  the  equa- 
tion by  5x(x2  —  4),  the  lowest  common  denominator. 

102.  — ? I '^~ =  0 

2x  +  1        3x  +  2  ^  6x2  +  7x  +  2      "■ 


103. 


12x  -  5         3x  +  4         4x  -  5 


21  3(3x  +  1)  7 

104.      ^'  +3      ,  1         _         2x  -  1 


2(x'  -  8)       6(x  -  2)       3(x2  +  2x  +  7)' 
3Q 


466      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§309 
106.  _£_4.a:-l_a;»+a;-l 


106. 


X  ~  1  X  x'  —  X 

X x__  _  a:'  +  23:  —  2 

s+2       a;  +  3~a;'  +  5a;+6* 


107.  ~ 1 Z?_ L     ^^     =  0 

a;  -  2  ^  24(a!  +  2)  ^4  -  a;^ 

108.  (.  +  -!)= -^(a;+g  +  7  =  i 

Hint:  Let  a;  +  -  =  j/.    Then 

2/'  -  Y!/  +  7  =  *■ 

Solve  this  equation  for  y.    Place  a;  +  -  equal  to  each  value  found 

for  y  and  solve  the  resulting  equations  for  x.     There  are  in  all  four 
roots  of  the  given  equation. 

109.  a;«  -  35a;«  +  216  =  0.  ,,,       _i_  1      o  i   i 
Hint:  Let  a;' =  v.  111.  a:  +  -  =  2  +  2. 

110.  .^+  ^0^29.  ii2.?i__24     +1=0. 

a;"  a;       a;  —  2 

113.  (a;2  +  a;)2  =  12  +  ^{yfl  +  a;), 
a;  +  1       12(a  -  1) 


114.  1  + 
116. 


a;-l  a;  +  l 

Ca;-l)(a;-2)  _(a;  +  l)Ca;  +  2) 
X  -Z  »  +  3 


116.  Solve  oV  _  2fa2/  +  bV  =  1  for  J/i  considering  a,  &,  ^,  and  a;  as 
known  numbers. 

307.  The  Definitions  of  Exponents. 

(1)  n  a  positive  integer:  o"  =  aaa  .    .    .  to  n  factors. 

(2)  n   and    r   positive    integers:    a^'  =  '^ a  and  a"/'  =  (Vo)" 

=  v^. 

(3)  o»  =  1. 

(4)  n  any  number,  positive  or  negative,  integral  or  fractional: 
a-»  =  l/o». 

308.  The  Laws  of  Exponents.    For  n  and  r  any  numbers,  positive 
or  negative,  integral  or  fractional : 

(1)  a"o'    =  o"*',  or  law  for  multipUoation  and  division. 


REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     467 

(2)  (o")'  =  a",  or  law  for  involution. 

(3)  o'S"  =  (aft)",  or  distributive  law  of  exponents. 

Note:    The  student  must  distinguish  between  —  a"  and  (—  a)". 

Thus  -  8!^  =  -  2,  and  (-  8)!^ 2,  but  (-  ZY   =  9  and  -  3'  = 

-9. 

Exercises  1 

Use  the  definitions  of  exponents  (1),  (2),  (3),  (4)  §307,  and  the  laws 
of  exponents  (1),  (2),  (3),  §308,  and  find  the  results  of  the  indicated 
operations  in  the  following  exercises. 


1.  x"x". 

9.  x"  -T-  i«. 

17.  (a')3. 

2.  a'^'a"'. 

10.  a"  -f-  a". 

18.  (o*)». 

3.  a;i'»+ii». 

11.  o'»  ^  a». 

19.  (-o62)». 

4.  626»+s. 

12.  e»+'  -7-  e». 

20.  (o»2/«)s. 

6.  W+'m""'. 

13.  IC'+s  -=-  10'. 

21.  (ft")  2. 

6.  o"-»o»+». 

14.  n'+»  -f-  n'+3. 

22.  (-o»6'>'. 

7.  s»-'+V. 

15.  u""'^  -^  u""'. 

23.  (a«6»)'. 

8.  m^'^mr''. 

16.  aj'-f+i  ^-  a;'. 

24.  (r'»s»)''. 

-  ©'■ 

Exercises  2 

27.    (- 

Write  each  of  the  following  sixteen  expressions,  using  fractional 
exponents  in  place  of  radical  signs: 


1.  v^. 

5.  v^ 

9.  v^iT 

13.  V'o-5. 

2.  V^- 

6.  (v^=. 

10.  (>^3. 

14.  (-C^a  -  6)'. 

3.  Vc^. 

1.-^. 

11.  </x'. 

16.  -s/a?  -  6''. 

4.  v^ 

8.  {<fS)K 

12.  (^^^ 

16.  y/ia+hy. 

Find    the    numerical    value    of    each    of    the    following    sixteen 
expressions: 


17.  4*. 

21.  625* 

25.  81^. 

29.  256*. 

18.  27*. 

22.  64*. 

26.  125^. 

30.  64^. 

19.  9*. 

23.  216*. 

27.  32^. 

31.  512^ 

20.  lei. 

24.  16^. 

28.  81^. 

32.  128*. 

Write  each  of  the  following  expressions  in  two  ways,  using  radical 
signs  instead  of  fractional  exponents: 


468      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§309 

S3,  o*. 

37. 

n^.                  41.  r*. 

46.  ol. 

34.  1*. 

38. 

b*.                  42.  xX 

46.  6"^. 

36.  TO*. 

39. 

^.                  43.  r- 

n+1 

47.  a;  -  . 

36.  s*. 

40. 

&i                  44.  ol 
Exercises  3 

48.  0  '• 

Perform  the  indicated  operations  in  each  of 

the  following  examples 

by  means  of  the  laws  of  exponents. 

1.  a?  X  a*. 

a* 

X  a*  =  a«+*  =  a^^A  = 

=  o^. 

2.  x^  X  a;i 

4.  a;i  X  s*. 

1                   4 

6.  X  :•>  X  a  8n 

3.  x^  X  K*. 

5.  o*  X  o^. 
8.  (^  H-  a^. 

S              r 

7.  o»  Xai:. 

a^ 

^  at  =  a^t  =  a4*-M  . 

=  aA. 

9.  h^  4-  hi. 

11.  Sa^ftt  H-  4o26^. 

13.  6at  -=-  3a*. 

10.  nfi  -H  vi 

A. 

12.  9a*  H-  a*. 
15.   (a^)^. 

14.  a6T  -=-  o    6"C. 

(a^)  ^  =  o^  =  oT^. 

» 

16.  (o*)A. 

18.  (a*)^. 

20.  [(x»)f]f. 

17.  (h^^. 

19.  (a*)*. 

22.  (ata;i-2/t)i 

21.  (si^)r. 

(a^x^y^ 

J)*=(at)W)*(2/¥  = 

;  a*x*j/i''. 

23.  (o^fe*)*. 

26.  (36a*a;22/')*. 

27.  (32x%4)*. 

24.   (,adi)i. 

26.  (a^a;^2/*)». 

28.  (^a'6'c)*. 

-  (i)*- 

/aiy_(af)i       a* 

VftV         (6*)^       6i 

-(?; 

-(i)' 

-(^)' 

"■©• 

A'\* 

-m 

36.  (a*  +  at  -f  l)(o*  -t-  a  -  cji). 


§309]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     469 

We  arrange  the  work  thus: 

J  +  J  +  1 
a''  +  a    —  gi 

ai  +J  +  a 

_  of  _  o  _  ai 


a'  +  2ai  +  a*        -  a^ 

37.  (x  +  22/4  +  32/*).(i  -  2yi  +  32/*). 

38.  (X*  +  yh(.x^  -  yh- 

39.  (,J  -  Sah^  +  4:ah  -  ah^-)(ai  -  2o*6*). 

3  i_  1  ^  ^ 

40.  (a»  -  20"+ 3o")(2a»  -  a"). 

Exercises  4 
Find  the  numerical  value  of  each  of  the  following; 

1.  2-1.  4.  10-5.  7.  2-\  10.  1024-*. 

2.  4-2.  5.  l-»  8.  16-".  11.  512-4. 

3.  (-2)-».  6.  2-2.  9.  81-4.  12.  625-*. 

1  5  5-2  16-i 

13.  i-  16.  (:r^,-       1,7. —^-  19.  ^3r- 

9     .  1-8  32-4  7-1 

14.  J-.  16.  I^i-  18.  ^^iir-  20.  — ^• 
3-2                         8  1                          21  49-1 

Write  each  of  the  following  expressions  without  using  negative 
exponents: 

21.  x-K  25.  5a-'.  29.  (a;  +  y)-\      33.  2o«a;-^-4. 

22.  x'y-K  26.30-^6-4.  30.  (- x)-^         34.  (- a^)-^ 

„o      1  „„     2a-2  „,      X*  „^      a-ibi 

23.  ^Ti-  27.  „,.  _,-  31.  -^-  35. n"^- 

„.    ■mr'  „„     0^6-*  „„    3(rl6-i  „„     3a26-2c-'' 

24-  -^-  28.     _■  _,■  32.  ^ — 5 —  36.  g^,.,,,^,.- 

a; "  a-3^-5  so-fj,  5o  ^b  =c  * 

Write  each  of  the  following  expressions  in  one  line: 

37  (1.         39  -^.        41  ^^-      43  ''"y"  - 

38    1.  40    ^^-^.  42    '^E^yll.  44     J??i 

38-  a'  f  •  3a-2r»  *2-     u>z-^  **"  ^=F^ 

16(a+  b)-'c4.  ^  ,  1   ,  1,  A-. 

"•  (a -  6)-lc-«  x3  +  x=i  +  X  +  a-i 


470      ELEMENTARY  MATHEMATICAL  ANALYSIS       [§309 

Exercises  6 

Perform  the  indicated  operations  in  each  of  the  following  by  means 
of  the  laws  of  exponents. 

1.  o«  X  o"".  4.  8a-*  X  3a'.  7.  m"*  X  m"*. 

2.  r"  X  r-i».  6.  m-i  X  tt*.  8.  Sax-'  X  kbx\ 

3.  c->  -r  c-K  6.  a;=  -=-  a;-".  9.  o-»6-»  -r-  ab"'. 

10,  (-  7o-»6-»)(-4oi'b-')(a-%''a;-'). 

11.  (2o*6-*)(a-*6*-  |o*b*  +  ah-^). 

12.  7a-»b-V-'-i-  80-26-%- 

13.  S6a;5J/-'^4  -^  Tai-iy-'a-*, 

14.  18o-i&lc-5  -=-  6a*bV«. 
16.  Cai'j^-^ai  -r  2a;-^3/iz-i. 

16.  (a-')!. 

17.  (o-s)-". 
18  (a«)-«. 

19.  (7i*)-». 

20.  (r-iyi. 

21.  (c-')^ 

22.  (obc)-*. 

33.   (bj. 


42.  (a2a;-i+3a=x-i')(4o-i  -  5a;-i  +  6ax-^) 
4a-i  -  5a;-i  +  &ax-' 

aH-^  +  3a'a;-' 

4ox-i  -  5a"a;-2  +  6o»a;-» 

12a'a;-'  -  15a»a;-°  +  18a«z-« 
4aa;-i  +  7a2a:-2  -    9a'a;-=  +  18a*a;-* 

43.  (2a;-*  -  3a;  +  4a;*)  (a;-?  -  2a;-*  +  3a;-*). 

44.  (a;-*  -  2a;-*^  +  y^){x-^  -  y^). 
46.  (3a;*  -  |x*  +  4)  X  2a;-*. 

46.  (a;-^  +  x'h  +  l)(a;"*  -  1). 

47.  (a;-*+  3/-=)  (a;-*  -  y-^). 

48.  {x^y  +  yh{x^  -  y-*). 


§309]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     471 

49.  (2o*-  3axi){3a-i  .+  2a;-*) (4a*a;*  +  9o-M). 
60.  (x-^  -  x-iyi  +  x-iy  -  y^)  -■  (a;-*  -  j/*). 

x~^  —  y^)x-^  —  x-^y'  +  x-'y  —  y'(3r^  +  y 
x-^  —  x-hji 

x'^y  —  y' 
x-^y  —y' 
Bl.  (x-'  +  2x-°-  -  Sx-i)  -i-  (x-\+  3s-i). 

309.  Reduction  of  Surds  or  Radicals. 

1.  //  any  factor  of  the  number  under  thi  radical  sign  is  an  exact 
■power  of  the  indicated  root,  the  root  of  that  factor  may  he  extracted  and 
written  as  the  coefficient  of  the  surd,  while  the  other  factors  are  left 
under  the  radical  sign. 


(1)  Thus, 

VS  =  V4  X  2 

=  VW2 

=  2v^ 

(2)  Also, 

</81  =  V27  X  3 

=  V27i/3 

=  3i/3 

(3)  Also, 

Vl&ax*  =  i/Sx'  X  2ax 

=  VSx'V2ax 

=  2x^2ax 

2.  The  expression  under  the  radical  sign  of  any  surd  can  always  he 
made  integral. 


(1)  Thus 


(2)  Also 


30 


(2 
3 

-'4x\' 

=  J—  X  18 
\27 

=  iVl8 
3 

(7. 

i 

*/7        2 
=  Vi  X  2  = 

^i^x" 

= 

\VT. 

472         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§309 
3.  W&  may  change  the  index  of  some  surds  in  the  following  manner: 
(1)  Thus,  a/I  =  vVi 

=  V2 
1,2)  Also,  VlOOO  =  -s/^^/W^Q 

=  Vio 

(3)  Also,  V2563^  =  v/VilPas 

\ 

=  v^lGca' 
A  surd  is  in  its  simplest  form  when  (1)  no  factor  of  the  expression 
under  the  radical  sign  is  a  perfect  power  of  the  required  root,  (2)  the 
expression  under  the  radical  sign  is  integral,  (3)  the  index  of  the  surd  is 
the  lowest  possible. 

Methods  of  making  the  different  reductions  required  by  this  defini- 
tion have  already  been  explained.     We  give  a  few  examples. 


(1)  Simplify  ^^^^ 


(2)  Simplify^—. 


a=  'a  1    3,_--  - 


*/400  /20      1      ,  2     ^__ 


(3)  Simplifying-^. 


5  « /512       5    /"S  h  .„ - 


2  \  125       2 ' 
f4)  Simplify  3\/2  +  2^-  +  Vs. 


3^2  +  2-/-  +  VS  =  3V2  +  \/2  +  2^2 
=  6V2. 


§318]     REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     473 

In  any  piece  of  work  it  is  usually  expected  that  all  the  surds  will 
finally  be  left  in  their  simplest  form. 

Exercises 
Reduce  each  of  the  following  surds  to  its  simplest  form : 

5.  ^  —  7.  ^  . 

5  \4  \81  \b'x^ 


\3  \27  \12  \        x' 


9.  Simplify  V^  +  AV^  +  6VS- 

10.  Simplify  1  +  Vs  +  V2  -  V27  -  Vl2  +  Vl5. 

11.  Sunplify   ■i/21  +  7v^  X  "^21  -  7V5. 

12.  Find  the  value  of  a;^  -  6a;  +  7  if  a;  =  3  -  \/3. 

13.  Find  the  value  when  x  =  \/3  of  the  expression 

2a;-  1    _   2a;  +  1 
(a;-  1)2        (x  +  1)2' 

14.  Find  the  value  of 

(35VT0  +  77\/2  +  63-v/3)(vT^  +  V2  +  \/3). 

Solve  and  check  each  of  the  following  equations: 

15.  vT+4  =  4.  22.  Vx  -  Vx -5  =  \/5. 

16.  ■v/2a;  +  6  =  4.  23.  Va;  -  7  =  V a;  -  14  +  1. 

17.  VlOa;  +  16  =  5^ 24.  Vx  -  7  =  -y/x  +  1  -  2. 

18.  V2x  +  7  =  VSx  -  2.  26.  x  =  7  -  Vx"  -  7. 

19.  14  +  -i^4x  -  40  =^10.  26.  Vx  +  20  -  Vx  -  1  -3=0. 

20.  Vl6x+9  =  4V'4x  -  3.  27.  -y/x  +  3  +  ■\/3x  -  2  =  7.  _ 

21.  V^-  +  X  =  f  +  Vx.  28.  V2x+  1+  Vx  -  3  =  2Vx. 

20x  , .  18 

29-  -7?^?=^  -  VlOx  -  9  =  -jrz :  +  9. 

VlOx  -  9  VlOx  -  9 

30    ^^^  _   Vi  +  1 


31. 


Vx  —  1 X  —  3 

Vx  +  Vx  —  3  _  3_ 

Vx  —  Vx  —  3        X  — : 


318.  Rationalizing  the  Benominator  of  a  Fraction. 
Illustration:  Rationalize  the  denominator  of 

V3  +  2  (V3  +  2)(V3+ V2)         3  +  2V3+V6  +  2V3 


V3-V2       (V3  -  V2)(V3  + V2) 


474       ELEMENTARY  MATHEMATICAL  ANALYSIS       [§318 


ExerciBes 

Rationalize  the  denominator  of  each  of  the  following : 

1  6  g   Vl^^^  +  1 
'  3  +  -v/s"  '  Vx  -2+2' 

2  VS  -  V2  ^    Vo  -  6  +  Va 
'  VB  +  -\/2  "  Va  —  b  —  Va 

g    2V2  +  3  g    Vl  +  g  -  Vl  -  g 

■  3\/2  +  2  '  Vl  +a  +  Vr^^ 

^    5>/2^+6  g  1 


3V2  -  6  ■  a;2  -  Vl  +  x' 

Vi  +  Va  '  1  +  Vi  -  x^ 


4  76       ELEMENTARY  MATHEMATICAL  ANALYSIS 


LOGARITHUB 


IO|il3l3l4lSl6l7l8l9lia 

3  U 

5  617  8  9  1 

10 

II 

12 

0000 

0043 

0086  0128 

0170 

0212 

0253 

0294 

0334 

0374 

4  8 

13 

12 

17 
16 

21  25 
20  24 

30  34  38 
28  32  37 

0414 
0792 

0453 
0828 

0492 
0864 

0531 
0899 

0569 
0934 

0607 
0969 

0645 
1004 

0682 
1038 

0719 

1072 

0755 
II06 

4  8 
4  7 
3  7 
3  7 

12 
II 
II 
10 

15 
IS 
14 
14 

19  23 
19  22 
18  21 
17  20 

27  31  3S 
26  30  33 
25  28  32 
24  27  31 

13 
14 

15 

l6 
17 

Is 

19 
20 

II39 
I46I 

I173 
1492 

1206 
1523 

1239 
ISS3 

1271 
1584 

1303 
1614 

I33S 
1644 

1367 
1673 

1399 
1703 

1430 
1732 

3  7 

3  I 
3  6 
3  6 

10 

10 

9 

9 

13 
12 
12 
12 

16  20 
16  19 
IS  18 
IS  17 

23  26  30 
22  25  29 
21  24  28 
20  23  26 

I76I 

1790 

1818 

1847 

1875 

1903 

193 1 

1959 

1987 

2014 

3  6 
3  5 

9 
8 

II 
II 

14  17 
14  16 

20  23  26 
19  22  25 

2041 
2304 

2068 
2330 

2095 
2355 

2122 
2380 

2148 
2405 

2175 
2430 

2201 

2455 

2227 
2480 

2253 
2504 

2279 
2529 

3  5 
3  5 
3  5 
2  5 

8 

8 
8 

7 

II 
10 
10 
10 

14  16 
13  15 
13  IS 

12  15 

19  22  24 
18  21  23 
18  20  23 
17  19  22 

2553 

2788 

2577 
2810 

2601 
2833 

2625 
2856 

2648 
2878 

2672 
2900 

2695 
2923 

2718 
2945 

2742 
2967 

2765 
2989 

2  5 
2  5 
2  4 
2  4 

7 
7 
7 
6 

9 
9 

12  14 
II  14 
II  13 
II  13 

16  19  21 
16  18  21 
16  18  20 

IS  17  19 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3I8I 

3201 

2  4 

6 

8 

II  13 

IS  17  19 

21 
22 
23 

24 

^1 

11 
29 

30 

31 
32 
33 

34 

II 

11 
39 

3222 
3424 
3617 

3243 

3263 
3464 
365s 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 
3560 

3747 

338s 

3579 
3766 

3404 
3598 
3784 

2  4 
2  4 
2  4 

6 
6 
6 

8 
8 

7 

10  12 
10  12 
9  II 

14  16  18 
14  15  17 

13  IS  17 

3802 

3979 
41SO 

3820 
3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 
4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 
4133 
4298 

2  4 
2  3 
2  3 

5 
5 
S 

7 
7 
7 

9  II 
9  10 
8  10 

12  14  16 
12  14  IS 
II  13  15 

4314 
4472 
4624 

4330 
4487 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4378 
4533 
4683 

4393 
4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

4456 
4609 
4757 

2  3 
2  3 
I  3 

5 
5 
4 

6 
6 
6 

8  9 
8  9 
7  9 

II  13  14 
II  12  14  ' 
10  12  13 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

I  3 

4 

6 

7  9 

10  II  13 

4914 
5051 
SI8S 

4928 
5065 
5198 

4942 
S079 
5211 

4955 
5092 
5224 

4969 
5105 
5237 

4983 
5119 
5250 

4997 
5132 
5263 

SOU 

5145 
5276 

5024 
5159 
5289 

5038 
5172 
5302 

13 
I  3 
I  3 

4 
4 
4 

6 
S 
5 

7  8 
6  8 

10  II  12 
9  II  12 
9  10  12 

S3IS 
S44I 
5563 

5328 
5453 
5575 

5340 
5465 
5587 

5353 
5478 
5599 

5366 
5490 
561I 

537.8 
5502 
5623 

5391 
5514 
563s 

5403 
5527 
5647 

5416 
5539 
5658 

5428 
5551 
5670 

I  3 
I  2 
I  2 

4 
4 
4 

5 
5 
5 

6  8 
6  7 

9  10  II 
9  10  II 
8  10  II 

5682 
5798 
591 1 

5694 
S809 
S922 

5705 
S821 
5933 

5717 
5832 
5944 

5729 
5843 
5955 

5740 
5855 
5966 

5752 
5866 
5977 

5763 
5877 
5988 

5775 
5888 
5999 

5786 
5899 
6010 

I  2 
I  2 
I  2 

3 
3 
3 

5 
5 

4 

6  7 

6  7 
S  7 

8  9  10 
8  9  10 
8  9  10 

40 

41 
42 
43 

6021 

6031 

6042 

6053 

6064 

6075 

608  s 

6096 

6107 

6117 

I  2 

3 

4 

S  6 

8  9  10 

6128 
6232 
633s 

5138 
5243 
634s 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
632s 
6425 

I  2 
I    2 
I  2 

3 
3 
3 

4 
4 
4 

5  6 

789 
789 
789 

44 

643s 
6532 
6628 

6444 

6454 
6551 
6646 

6464 
6561 
6656 

6474 
6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 
6693 

6513 
6609 
6702 

6522 
6618 
6712 

I  2 
I  2 
I  2 

3 
3 
3 

4 
4 
4 

S  6 

789 
7  8  9 

7  7   8 

49 

6721 
6812 
6903 

6730 
6821 
6911 

6739 
6830 
6920 

6749 
6839 
6928 

6758 
6848 
69J7 

6767 
6857 
6946 

677616785 
6866  6875 
6955  6964 

6794 
6884 
6972 

6803 
6893 
6981 

I  2 

1  2 
1  2 

3 
3 
3 

4 
4 
4 

5  5 
4  5 
4  5 

678 
678 
678 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042  7OS0I7059 

7067 

I  2 

3 

3 

4  sl  6  7  8 

REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     477 


LoGAEITHMS 


lo  ii|2  1  3(4ISl6|7\8  I9I123U  5  617  89I 

51 

52 

53 

7076:7084 
7160  7168 
72437251 

7093 
7177 
7259 

7101  7110 
718s  7193 
7267  7275 

7118 
7202 
7284 

7126 
7210 
7292 

7135 
7218 
7300 

7143 
7226 
7308 

7152 
7235 
7316 

I  2   3 
12   2 
12   2 

3  4  5 
3  4  5 
3  4  5 

678 
677 
6  6 

54 
55 

73247332 
7404  7412 

7340 
7419 

7348  7356 
7427  7435 

7364 
7443 

7372 
7451 

7380 
7459 

7388 
7466 

7396 
7474 

12   2 

12   2 

3  4  5 
3  4  5 

667 

5  6  7 

56 

7482  7490 
7SS97S66 
7634  7642 

7497 
7574 
7649 

7505  7513 
7582  7589 
7657  7664 

7520 
7597 
7672 

7528 
7604 
7679 

7536 
7612 
7686 

7543 
7619 
7694 

7551 
7627 
7701 

12   2 
12   2 
112 

3  4  5 
3  4  5 
3  4  4 

5  6  7 
5  6  7 
567 

61 
62 
64 

7709  7716 
7782  7789 
7853  7860 

7723 
7796 
7868 

7731  7738 
7803  7810 
7875  7882 

7745 
7818 
7889 

7752 
782s 
7896 

7760 
7832 
7903 

7767 
7839 
7910 

7774 
7846 
7917 

112 
112 
112 

3  4  4 
3  4  4 
3  4  4 

5  6  7 
566 
S  6  6 

7924 
7993 
8062 

7931 
8000 
8069 

7938 
8007 
8075 

7945  7952 
80I4'802I 
8082  8089 

7959 
8028 
8096 

7966  7973 
8035  8041 
8l02;8l09 

7980 
804S 
8116 

.7987 
8055 
8122 

I  Z   2 
112 
112 

3  3  4 
3  3  4 
3  3  4 

566 
556 
5  S  6 

6s 

8129 

8136 

8142 

8149  8156 

8162 

8169 

8176 

8182 

8189 

112 

3  3  4 

5  56 

66 

? 

69 
70 
71 

72 
73 
74 

8I9S 
8261 
832s 

8202 
8267 
8331 

8209 
8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 
8363 

8241 
8306 
8370 

8248 
8312 
8376 

82S4 
8319 
8382 

112 
112 
112 

3  3  4 
3  3  4 
3  3  4 

5  5  6 
5  5  6 
456 

8388 
84s  I 
8513 

8395 
8457 
8519 

8401 
8463 
8525 

8407 
8470 
8531 

8414 
8476 
8537 

8420 
8482 
8543 

8426 
8488 
8549 

8432 
8494 
8555 

8439 
Ssoo 
8561 

8445 
8506 
8567 

112 
112 
112 

234 
234 
234 

4  5  6 
456 
4  5  5 

5f" 
8633 
8692 

8579 
8639 
8698 

!l*5 
8645 

8704 

8591 
8651 
8710 

till 
8716 

8603 
8663 
8722 

8609 
8669 
8727 

8615 
867s 
8733 

8621 
8681 
8739 

8627 
8686 
8745 

Z  I   2 
112 
112 

234 
234 
234 

4  5  5 
4  5  5 
4  5  5 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

112 

233 

4  5  5 

76 
11 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

882s 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8904 
8960 

8854 
8910 
8965 

8859 
8915 
8971 

112 
112 
112 

233 
233 
233 

4  5  5 
4  4  5 
4  4  5 

81 

83 
84 

8976 
9031 
908s 

8982 
9036 
9090 

8987 
9042 
9096 

8993 
9047 
910I 

8998 
9053 
9106 

9004 
9058 
9112 

9009  901S 
9063 '9069 
9117J9122 

9020 
9074 
9128 

902s 
9079 
9133 

112 
112 
112 

233 
233 
233 

4  4  5 
4  4  5 
4  4  5 

9138 
9I9I 
9243 

9143 
9196 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9159 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9175 
9227 
9279 

9180 
9232 
9284 

9186 
9238 
9289 

112 
112 
112 

233 
233 
233 

4  4  5 
4  4  5 
4  4  5 

85 
86 

9294  9299 

9304 

9309 

93IS 

9320 

9325 

9330 

9335 

9340 

112 

233 

4  4  5 

9345 
939S 

9445 

9350 
9400 
9450 

9355 
940s 
9455 

9360 
9410 
9460 

9365 
9415 
946s 

9370 
9420 
9469 

9375 
9425 
9474 

9380 
9430 
9479 

9385 
9435 
9484 

9390 
9440 
9489 

112 
Oil 
0  1   1 

233 
223 
223 

4  4  5 
3  4  4 
3  4  4 

89 
90 
91 

9494 
9542 
9590 

9499 
9547 
9595 

9504 
9552 
9600 

9509 
9SS7 
960s 

9513 
9562 
9609 

9518 
9566 
9614 

9523  9528 
9571  9576 
9619  9624 

9533 
9581 
9628 

9538 
9586 
9633 

Oil 
0  1   1 
Oil 

223 
223 
223 

3  4  4 
3  4  4 
3  4  4 

92 
93 
94 

9638 
968s 
9731 

9643 
9689 
9736 

9647 
9694 
9741 

9652 
9699 
9745 

9657 
9703 
9750 

9661 
9708 
97S4 

9666  9671 
9713  9717 
9759  9763 

9675 
9722 
9768 

9680 
9727 
9773 

Oil 
0  1   1 
0  1   I 

223 
223 
223 

3  4  4 
3  4  4 
3  4  4 

95 
96 

9777 

9782 

9786 

9791 

9795 

9800 

980s  9809 

9814 

9818 

Oil 

223 

3  4  4 

9823 
9868 
9912 

9827 
9872 
9917 

9832 
9877 
9921 

9836 
9881 
9926 

9841 
9886 
9930 

984s 
9890 
9934 

9850  9854 
9894  9899 
9939  9943 

9859 
9903 
9948 

9863 
9908 
9952 

0  1   I 

oil 

0  1   I 

223 
223 
223 

3  4  4 
3  4  4 
3  4  4 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983  9987 

9991 

9996 

oil 

223 

3  3  4 

The  copyright  of  that  portion  of  the  above  table  which  gives  the  logarithms  of 
numbers  from  1000  to  2000  is  the  property  of  Messrs.  Macmillan  and  Company, 
limited,  who,  however,  have  authorised  the  use  of  the  form  in  any  reprint  pub- 
lished for  educational  purposes. 


478       ELEMENTARY  MATHEMATICAL  ANALYSIS 
Logarithms  of  Tbiqonometric  Fttnctions 


o      / 

log  sin 

d 

log  tan 

dc 

log  cot 

log  cos 

' 

S 

T 

0.0000 

0  90 

0 

0      w 

10 

7.4637 

3011 

7-4637 

3011 

2.5363 

0.0000 

so 

10 

6.4637 

6.4637 

20 

7.7648 

1760 

12S0 
969 

7.7648 

1 761 
1249 
969 

2.2352 

0.0000 

40 

20 

6.4637 

6.4637 

30 

7.9408 

7.9409 

2.0591 

0.0000 

30 

30 

6.4637 

6.4637 

40 

8.0658 

8.0658 

1.9342 

0 . 0000 

20 

40 

6.4637 

6.4637 

SO 

8.1627 

792 
669 
580 

8.1627 

792 
670 
s8o 

I . 8373 

0.000b 

10 

50 

6.4637 

6.4638 

I       0 

8.2419 

8.2419 

1.7581 

9.9999 

0  89 

60 

6.4637 

6.4638 

10 

8.3088 

8 . 3089 

1.6911 

9.9999 

SO 

70 

6.4637 

6.4638 

20 

8.3668 

SII 
458 
413 

8.3669 

5" 
457 
41S 

1.6331 

9.9999 

40 

80 

6.4637 

6.4638 

30 

8.4179 

8.41S1 

I.S8I9 

9.9999 

30 

90 

6.4637 

6.4638 

40 

8.4637 

8 . 4638 

1.5362 

9.9998 

20 

100 

6.4637 

6.4638 

so 

8.S0S0 

378 
348 
321 

8.S053 

378 
348 
322 

1.4947 

1. 4569 

9.9998 

10 

110 

6.4637 

6.4639 

2       0 

8.S428 

8.S43I 

9.9997 

0  88 

120 

6 . 4636 

6.4639 

10 

8.S776 

8.5779 

1.4221 

9.9997 

SO 

130 

6.4636 

6.4639 

20 

8 . 6097 

300 
280 

8.6101 

300 
281 
263 

1.3899 

9.9996 

40 

140 

6 . 4636 

6.4640 

30 

8.5397 

8.6401 
8.6682 

I.3S99 

9.9996 

30 

ISO 

6 . 4636 

6.4640 

40 

8.6677 

263 

1.3318 

99995 

20 

160 

6.46J6 

6 .  4640 

so 

8 . 6940 

248 
235 
222 

8.694s 

249 
235 
223 

1.3OSS 

9.9995 

10 

170 

6.463s 

6.4641 

3      0 

8.7188 

8.7194 

1.2806 

9.9994 

0  87 

180 

6.463s 

6.4641 

10 

8.7423 

8.7429 

I.2S71 

9.9993 

SO 

190 

6.4635 

6.4642 

20 

8.764s 

212 

8.7652 

213 

1 .  2348 

9.9993 

40 

200 

6.463s 

6.464a 

30 

8.7857 

8.786s 

I. 213s 

9.9992 

30 

210 

6.463s 

6.4643 

40 

8.8059 

192 

8 .  8067 

194 

1.1933 

9.9991 

20 

220 

6.4634 

6.4643 

so 

8.82SI 

18s 
177 
170 

8.8261 

178 
171 

1.1739 

9.9990 

10 

230 

6.4634 

6.4644 

4     0 

8 . 8436 
8.8613 

8 . 8446 
8.8624 

I.1SS4 

9.9989 

0  86 

240 

6.4634 

6.4644 

10 

1.1376 

9.9989 

SO 

250 

6.4633 

6.4645 

20 

8.8783 

IS8 
152 

8.879s 

IS8 
154 

1.1205 

9.9988 

40 

260 

6.4633 

6 .  4646 

30 

8.8946 

8. 8960 

1 . 1040 

9.9987 

30 

270 

6.4633 

6 .  4646 

40 

8.9104 

8.9118 

1.0882 

9.9986 

20 

280 

6.4632 

6.4647 

so 

8.9256 

147 

8.9272 

148 

1.0728 

9.998s 

10 

290 

6.4632 

6.4648 

5 

0 

8.9403 

8.9420 

1.0580 

9.9983 

0  8s 

300 

6.4631 

6 . 4649 

1  log  COB   1    d     1  log  cot^  I    dc    1  log  tan 

Hog  Sin   1 '     °  1           1               1             1 

113 

142 

138 

137 

13S 

134 

130 

129 

19 

7 

12s    1 

23 

1S2 

119 

117 

lis 

lU 

1 

U.3 

14.2 

13.8 

13. 

7   13.5 

13.4 

13. 

»    12.9 

12 

.7 

12.6 

12.3 

12.2 

11. 

)  11.7 

11.5 

11.4 

2 

28.6 

28.4 

27.6 

27. 

4   27.0 

26.8 

26. 

a   25.8 

2S 

.4 

25.0 

!4.6 

24.4 

23. 

i  23.4 

23.0 

22.8 

3 

42.9 

42.6 

41.4 

41. 

1   40.5 

40.2 

39. 

D   38.7 

3E 

.1 

37.5 

16.9 

36.6 

35. 

J  35.1 

34.5 

34.2 

1 

57.2 

56.8 

55.2 

54. 

8   54.0 

53.6 

52. 

9   51.6 

6C 

.8 

50.0  ' 

19.2 

48.8 

47. 

i  46.8 

46.0 

45. 6 

S 

71.5 

71.0 

69.0 

68. 

6   67.5 

67.0 

65. 

a  64.5 

63 

.5 

62.5 

H.5 

61.0 

59. 

>  58.6 

67.6 

57.0 

6 

85.8 

85.2 

82.8 

82. 

2   81.0 

80.4 

78. 

D   77.4 

7( 

.2 

75.0 

r3.8 

73.2 

71. 

I  70.2 

69.0 

68.4 

7 

100.1 

99.4 

96.6 

95. 

9   94.5 

93.8 

91. 

D   90.3 

8! 

.9 

87.6 

36.1 

85.4 

83. 

i  81.9 

80.5 

79.8 

8 

114.4 

113.6 

110.4 

109. 

6  108.0 

107.2 

104. 

0  103.2 

101 

.6 

100.0 

98.4 

97,6 

95. 

!  93.6 

92.0 

91.2 

128.7 

127.8 

124.2 

123. 

3  121.5 

120.6 

117. 

D  116.1 

IM 

.3 

112. 5  1 

10.7 

109.8 

107. 

1105.3 

103.5 

102.6 

Formulas  for  using 

Table  directly 

%  !  log  sin  *  =  log  I*  +  S                     °io 

log  cos  X  =  log  (90  -  *)'  +  S 
log  cot  *  =  log  (90  -  *)'  +  T 
log  tan  X  =  colog  (90  —  xy  +  co  T 

V    log  tan  X  =  log  I*  +  T                    " 
«  [  log  cot  *  =  colog  I*  +  CO  T            ^ 

REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     479 
Logarithms  of  Tbigonometeic  Functions 


log  sin 


log  tan 


dc 


log  cot 


log    COS 


pp 


so 


so 


SO 


8.9403 
8.9S4S 

8. 9682 
8.9816 
8. 9945 

9 . 0070 
9.0192 
9.031I 

9..  0426 
9.0S39 
9 .  0648 

9.07SS 
9.0859 
9 .  0961 

9.1060 
9.1157 
9.1252 

9.1345 
9.1436 
9.1525 

9.1612 
9.1697 
9.1781 

9.1863 
9-1943 
9.2022 

9.2100 
9.2176 
9.2251 

9.2324 
9.2397 


log    COS 


142 
137 

134 

129 

125 

122 
119 

IIS 

113 

109 
107 

104 

102 
99 

97 
95 
93 

9f 
89 

8t 

85 
84 
82 

80 
79 
78 

76 
75 
73 


8.9420 
8.9563 

8.9701 
8.9836 
8 . 9966 

9.0093 
9.0216 
9.0336 

9.0453 
9.0567 
9.0678 

9.0786 
9.0891 
9.0995 

9 . 1096 
9.1194 
9.1291 

9.1385 
9.1478 
9.1569 

0.1658 
9. 1745 
9.1S31 

9.1915 
9.1907 
9.2078 

9.2158 
9.2236 
9.2313 

S-2389 
9.2463 


log  cot 


143 
138 

13s 
130 

127 

123 

120 
117 

114 
111 
108 

105 
104 
lOI 

98 

97 
94 

93 
91 
89 

87 
86 
84 

82 
81 
80 

78 
77 
76 


1.0580 
1.0437 

1.0299 
1.0164 
1.0034 

0.9907 
0.9784 
0.9664 

0.9547 
0.9433 
0.9322 

0.9214 
0.9109 
0.9005 

0.8904 
9.8806 
0.8709 

0.8615 
0.8522 
0.8431 

o .  8342 
0.8255 
0.8169 

0.8085 
0.8003 
0.7922 

0.7842 
0.7764 
0.7687 

0.7611 
0.7537 


9.9983 
9.9982 

9.9981 
g.9980 
9.9979 

9.9977 
9.9976 
9.9975 

9.9973 
9.9972 
9.9971 

9.9969 
9.9968 
9.9966 

9.9964 
9.9963 
9.9961 

9.9959 
9-9958 
9.9956 

9.9954 
9.9952 
9.9950 

9.9948 
9.9946 
9.9944 

9.9942 
9.9940 
9.9938 


9.9936  10 
9.9934    O  80 


o  8S 


10 

0  84 
50 

40 
30 
20 


o  83 


40 
30 
20 

10 

0  82 
SO 

40 
30 
20 


0  81 


dc 


log  tan 


log  sin 


"3 

II. 3 
22.6 
33.9 

III 
II. I 

22.2 
33.3 

45.2 
67i8 

44.4 

55. S 
66.6 

79.1 
90.4 
101.7 

77.7 
88.8 
99.9 

108 

10.8 
21.6 
32.4 

107 
10.7 
21.4 
32.1 

43.2 

64.8 

42.8 
53. 5 
64.2 

75.6 
86.4 
97-2 

74.9 
85.6 
96.3 

104 
10.4 
20.8 
31.2 

102 
10.2 
20.4 
30.6 

41.6 

52.0 

62.4 

40.8 
51.0 
61.2 

72.8 
83.2 

93.6 

il'.6 
91.0 

109 

10. 9 
21.8 
32-7 

43-6 
54-5 
65.4 

76.3 
87.2 
98.1 

lOS 
10.5 
21.0 
3I-S 

42 -0 
525 

63.0 
73.  S 

84.0 

94-S 

lOI 

10. 1 

20.2 

30.3 

40.4 
50.5 
60.6 

70.7 
80.8 
90.9 


94 

9.4 
18.8 
28.2 

37.6 
47.0 
58.4 

65.8' 
75.2 
84.6! 


18.6 
27.9 

37.2 
46.9 
SS.S 

65.1 

74.4 
83.7 


91 

9.1 
18.2 
27.3 

36.4 
46.5 
54. 6 

63.7 
72.8 
81.9 


89 

8.1 
17.8 
26.7 

35.6 
44.5 
53.4 

62. 

71.2 

80.1 


87 

8.7 
17.4 
26.1 

31.8 
43.5 
92.2 

60.9 
69.6 
78.3 


86 

8.6 
17.2 
25.8 

34.4 
43.0 
91.6 


77.4 


86 

8.5 
17.0 
15.5 

34.0 
42.9 
91.0 
59.9 
68.0 
76.9 


84 

8.4 
16.8 
25 

33 

42.0 

50.4 

98.8 
67.2 
79.6 


16.4 
24.6 

32.8 
41.0 
49.2 

67.4 
65.6 
73.8 


81 

8.1 
16.2 
2.34 

32.4 

40 

48.6 

56.7 
64.8 
72.9 


79  I  78 

7.9  7.8 
19.815.6 
23.7,23.4 

31.631. 
39.9;39. 

47.446. 


0 

8 

95. 3^94. 6 
63.262.4 
71.1170.2 


99 

9.9 
19-8 
29-7 
39.6 

49. S 

S9.4 
69.3 
79-2 
89 -I 

77 

7.7 
15.4 
23.1 

30.1 
38.! 
46.: 


98 

9.8 
19.6 
29.4 

39.2 

49 

58.8 

68.6 
78.4 


97 

9-7 
19.4 
29.1 

38.8 


95 

9.5 
19. 
28.5 
38. 0 


48.5  47. S 

57. o 


58.2 

67.9 
77-6 
87.3 


66. s 
76.0 
85. S 


76 1  78  I  74 
7.6  7.5  7.4 
19.2:19.014.8 

1.5 


61 


78 

7.3 
14.6 
21.9 


l.4'30. 
1.037, 
i.649 


.4167 


0'29. 629.2 
537.0:36.5 
.0j44.4p.g 

ffsi.ffgi.i 

.099.298.4 
.9|68.6i65.7 


Formulas  for  usine  Table  inversely 


Ilog  *'■=  log  sin  X  —  S 
log  «'  =  log  tan  X  —  T 
colog  a/  =  log  cot  X  —  CO  T 


log  (90  —  x)'  ' 
log  (90  -  x)'  • 
colog  (90  —  xy 


'  log 
■■  log 
■  log 


COS 

cot 

tan 


X  — 

X  — 


S 
T 
CO  T 


480       ELEMENTARY  MATHEMATICAL  ANALYSIS 
Logarithms  op  Trigonometbic  Functions 


o   / 

log  sin 

d 

log  tan 

dc 

log  cot 

log  cos 

d 

pp 

10  0 

9   2397 

71 
70 

9-2463 

0-7S37 

9.9934 

3 
2 

0 

80 

73 

71 

10 

9.246S 

9-2536 

73 
73 

0.7464 

9.9931 

SO 

I 
2 

7-3 
14.6 

7-1 
14.2 

20 

9.2538 

68 

9-2609 

71 
70 
69 

0.7391 

9-9929 

2 

40 

3 

21.9 

21-3 

30 
40 

9 .  2606 
9.2674 

68 
66 

9-2680 
9.2750 

0.7320 
O.72SO 

9-9927 
9.9924 

3 
2 

30 
20 

4 

5 

29.2 
36. 5 

28.4 
35. S 

50 

9.2740 

66 

9.2819 

68 

0.7181 

9.9922 

3 
2 

10 

6 

43.8 

42.6 

II  0 

9.2806 

54 
64 

9.2887 

66 

67 

0.7113 

9.9919 

0 

79 

7 

SI.l 

4?Z 

10 

9.2870 

9.2953 

0.7047 

9.9917 

3 

50 

8 
9 

S8.4 
6S.7 

S6.8 
63.9 

20 

9.2934 

53 

61 

61 

9.3020 

65 
63 

0.6980 

9-9914 

2 

40 

70 

7.0 
14.0 

59 

6.9 

13.8 

30 

9.2997 

9.3085 

0.691S 

9.9912 

3 

2 

30 

40 

9-3058 

9.3149 

0.6851 

9.9909 

20 

2 

so 

9.3119 

60 
59 
58 

9.3212 

63 

61 
61 

0.6788 

9-9907 

3 
3  . 

2 

10 

3 

21.0 

20.7 

12  0 

9.3179 

9.3275 

0.672s 

9.9904 

0 

78 

4 

28.0 

27.6 

10 

9.3238 

9-3336 

0 . 6664 

9-9901 

SO 

35.0 
42.0 

34-S 
41.4 

20 

9.3296 

57 

9.3397 

61 

0 . 6603 

9.9899 

3 
3 
3 

40 

7 
8 
9 

49.0 
56.0 
63.0 

48-3 
55-2 
62.1 

30 
40 

9.3353 
9.3410 

9.3458 
9-3517 

59 
59 

0.6542 
0.6483 

9.9896 
9.9893 

30 
20 

SO 

9.3466 

55 
54 
54 

9-3576 

58 

57 
57 

0.6424 

9.9890 

3 
3 
3 

10 

« 

68 

67 

13  0 

9.3521 

9-3634 

0.6366 

9.9887 

0 

77 

I 

6.8 

6.7 

10 

9.357s 

9-3691 

0.6309 

9.9884 

50 

2 
3 

13.6 
20.4 

13.4 
20.  I 

20 

9.3629 

53 
S2 
52 

9-3748 

56 
55 
5S 

0.6252 

9.9881 

3 
3 
3 

40 

4 
5 
6 

27.2 
34-0 
40.8 

26.8 

30 

9.3682 

9.3804 

0.6196 

9.9878 

30 

33.5 
40.2 

40 

9-3734 

9.3859 

0.6141 

9.987s 

20 

SO 

9-3786 

SI 

SO 

50 

9.3914 

54 
53 
S3 

0.6086 

9.9872 

3 
3 
3 

10 

7 

47.6 

46.9 

14  0 

9.3837 

9.3968 

0.6032 

9.9869 

0 

76 

8 

54.4 

53-6 

ID 

9.3887 

9.4021 

0.5979 

9 . 9866 

50 

9 

61.2 

60.3 

20 

9.3937 

49 
48 

9.4074 

S3 
SI 
52 

0.5926 

9.9863 

4 
3 
3 

40 

66 

6.6 

6.5 
13.0 
19. 5 

30 

9-3986 

9-4127 

0.5873 

9.9859 

30 

13.2 
19-8 

40 

9 -403s 

9.4178 

0.5822 

9.9856 

20 

3 

SO 

9.4083 

47 

9.4230 

51 

0.5770 

9. 9853 

4 

10 

4 

26..^ 

26.0 

IS  0 

9-4130 

9.4281 

0.S7I9 

9.9849 

0 

75 

5 

33-0 

32.5 

6 

7 

39-6 
46.2 

39.0 
45.5 

log  COS 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

° 

8 
9 

52.8 
59.4 

SO 

52.0 
58.5 

48  1  47 

64 

fS 

6 

I 

6 

0 

59 

S8 

57 

S6, 

55 

54 

53 

52 

31  1 

I 

6.4 

6.3 

6 

.1 

6 

.0 

S.9 

5.8 

5.7 

5.6 

S-S 

5-4 

5-3 

s 

2 

S.I 

5.0 

4-8!  4.7 

2 

12.8 

12.6 

12 

.2 

12 

.0 

11.8 

II. 6 

II-4 

II. 2 

I.O 

10.8 

10.6 

TO 

4 

10.2  1 

0.0 

9.6  9.4 

3 

19.2 

18.9 

18 

■  3 

18 

.0 

17.7 

17-4 

17-I 

16.8 

6.S 

16.2 

15-9 

15 

6 

IS. 3  I 

5.0  I 

4-4 

14.1 

4 

2S.6 

2S.2 

24 

•4 

24 

.0 

23.6 

23-2 

22.8 

22.4  . 

>2.0 

21.6 

21-2 

20 

8 

20.4! 

0.0  ) 

9-2 

18.8 

S 

32.0 

3I-S 

30 

■  5 

30 

.0:29.5 

29.0 

28. s 

28.0  i 

7-S 

27.0 

26.5 

26 

0 

25.5: 

S.o: 

4.0 

23-5 

6 

38.4 

37.8 

36 

.6 

36 

-035-4 

34-8 

34-2 

33.6. 

i3-0 

32.4 

31.8 

31 

2 

30.6 : 

0.0  i 

8.8 

28.2 

7 

44.8 

44.1 

42 

.7 

42 

.041.3 

40-6 

39-9 

39-2  , 

i8-5 

37-8 

37-1 

36 

i 

3S.7: 

S-o; 

3-6 

32.9 

8 

SI. 2 

SO. 4 

48 

.8 

48 

.047.2 

46-4 

45-6 

44.8- 

t4-0 

43-2 

42-4 

41 

40.8  i 

io-o  ; 

8.4 

37.6 

9 

S7.6 

S6.7 

54 

-9 

54 

.o;s3.i 

S2.2 

SI-3 

50.4  ' 

J9-S 

48-6 

47.7 

46 

8 

45.9^ 

tS.Oi 

J3-2 

42.3 

KEViEW  OF  SECONDARY  SCHOOL  ALGEBRA     481 
Logarithms  op  Thigonomethic  Functions 


log  sin 


log  tan 


dc 


log  cot 


log  COS 


IS    o 


20 
30 
40 


i6    0 


i8    0 


20 
30 
40 


19    o 


9.4130 

9.4177 

9.4223 
9.4269 
9.4314 

9.43S9 
9.4403 
9-4447 

9-4491 
9-4533 
9.4576 

9.4618 
9.4659 
9.4700 

9-4741 
9.4781 
9.4821 

9.4861 
9.4900 
9-4939 

9.4977 
9-5015 
9-S052 

9.S090 
9.5126 
9.S163 

9.5199 
9-5235 
9.5270 

9-^306 
9-5341 


9.4281 
9.4331 

9.4381 
9.4430 
9.4479 

9.4527 
9-4575 
9.4622 

9.4669 
9-4716 
9-4762 


,4808 
4853 
14898 


9-4943 
9-4987 
9-5031 

9-5075 
9-5118 
9.5161 

9.5203 

9-5245 
9-5287 

9-5329 
9-5370 
9-5411 

9-5451 
9-5491 
9-5531 

9-5571 
9-S611 


0-5719 
0-5669 

0,5619 
0-5570 
0.5521 

0-5473 
0-5425 
0.5378 

0-5331 
0.5284 
O-S238 

0.5192 
0-S147 
0.5102 

0-5057 
0.5013 
o . 4969 

0.492s 
0.4882 
0.4839 

0.4797 
0.475s 
0.4713 

0.4671 
0.4630 
0.4589 

0.4549 
0.4509 
0.4469 

0.4429 
0.4389 


9-9849 
9.9846 

9.9843 
9.9839 
9.9836 

9.9832 
9.9828 
9.9825 

9.9821 
9.9817 
9-9814 

9-9S10 
9.9806 
9.9802 

9.9798 
9-9794 
9.9790 

9.9786 
9-9782 
9.9778 

9.9774 
9.9770 
9-9765 

9.9761 
9.9757 
9.9752 

9-9748 
9  9743 
9.9739 

9-9734 
9-9730 


0  75 


0  72 


50  71 


25.0 
30.0 

35-0 
40.0 
45 -Q 


48 

4.8 

9.6 

14.4 

ig.2 
24.0 
28.8 

33.6 
38.4 
43 


49 

4-9 
9.8 

14-7 


log  cos 


log  cot 


dc 


log  tan 


log  sin 


46 1 

4 

6 

0 

2 

13 

8 

18 

4 

2,1 

0 

27 

6 

,12 

2 

36 

8 

41 

4 

47 

4-7 
9-4 
14-1 

18.8 
23.  5 
28-2 

32.9 
37-6 
42-3 


45 

4-5 
9.0 
13.5 


31.5 
36.0 
40,5 


44 

13-2 

43 
12.9 

42 

4,2 

8-4 

12-6 

1 
2 
3 

41 

12.3 

40 
12.0 

39 

7.8 
1 1..  7 

3.8 

7.6 

11.4 

1 

2 
3 

37 

3-7 

7-4 

11- 1 

17-6 
22.0 
26.4 

17.2 

21. 5 
2S.8 

16.8 
21.0 
25.2 

4 
5 
6 

16.4 
20.5 
24.6 

16.0 
20.0 
24.0 

15.6 
19. 5 
23-4 

15.2 
19.0 
22.8 

4 
5 
6 

14.8 
18.5 
22.2 

30.8 
35.2 
39.6 

30.1 
34-4 
38.7 

29.4 
33-6 
37-8 

7 
8 
9 

28.7 
32.8 
36.9 

28.0 
32.0 
36.0 

27-3 
31-2 
3S-I 

26.6 
30.4 
34-2 

7 
8 
9 

25-9 
29.6 
33.3 

36 

3.6 

7.2 

10.8 

14-4 
18-O 
21.6 

25.2 
28.8 
32.4 


35 
35 

7-0 
10-5 

14.0 
17-5 


24- 5 
28.0 
31-5 


31 


482       ELEMENTARY  MATHEMATICAL  ANALYSIS 

LOQABITHMS  OF  TbIOONOMETRIC  FUNCTIONS 


■  1 

log  Sin 

d 

log  tan 

dc 

log  cot 

log  COS 

d 

PP 

ao  0 

9.S34I 

34 

34 

9.5611 

39 
39 

0.4389 

9.9730 

S 
4 

0  70 

10 

9-S37S 

9.5650 

0.4350 

9.9725 

50 

4 
0.4 

20 

9.S409 

34 
34 
33 

9.5689 

38 
38 

0.43" 

9,9721 

5 

40 

I 

30 
40 

9-S443 
9.S477 

0.5727 
9.5766 

0.4273 
0.4234 

9.9716 
9.9711 

5 
5 

30 

20 

2 
3 

0.8 
1.2 

so 

9-SSiO 

33 
33 
33 

9.5804 

38 
38 

0 .  4196 

9.9706 

4 
5 
5 

10 

4 

1.6 

31   0 

9.SS43 

9.5842 

0.4158 

9.9702 

0  69 

5 

2,0 

10 

9.SS76 

9.5879 

0.4121 

9.9697 

SO 

6 

2.4 

30 

9.S609 

32 
32 
31 

9.S9I7 

37 
37 
37 

0.4083 

9.9692 

5 
5 
5 

40 

7 

2.8 

30 

9.S641 

9.5954 

0.4046 

9.9687 

30 

8 

3.2 

40 

9.5673 

9.5991 

0.4009 

9.9682 

20 

9 

3.6 

so 

9.S704 

32 
31 
31 

9.6028 

35 
36 

0.3972 

9.9677 

5 
5 
6 

10 

aa  0 

9.5736 

9.6064 

0.3936 

9.9672 

0  68 

10 

9.5767 

9.6100 

0.3900 

9.9667 

SO 

I 
2 

5 
0.5 
1.0 

20 

9.5798 

30 
31 
30 

9.6136 

36 
36 
35 

0.3864 

9.9661 

5 
S 
5 

40 

30 

9.5828 

9.6172 

0.3828 

9.9656 

30 

3 

1.5 

40 

9.5859 

9.6208 

0.3792 

9.9651 

20 

4 
5 

2.0 

2.5 

SO 

9.5889 

30 
29 
30 

9.6243 

36 
35 

34 

0.3757 

9.9646 

6 

10 

23  o 

9.5919 

9.6279 

0.3721 

9.9640 

% 

0  67 

6 

3.0 

10 

9.5948 

9.6314 

0.3686 

9.9635 

50 

7 

3.5 

20 

9.5978 

29 
29 
29 

9.6348 

35 

34 
35 

0.3652 

9.9629 

1 

40 

8 

4.0 

30 

9.6007 

9.6383 

0.3617 

9.9624 

30 

9 

4.5 

40 

9.6036 

9.6417 

0.3583 

9.961:8 

5 

20 

so 

9.6065 

28 

9.6452 

34 
34 
33 

0.3548 

9.9613 

6 

10 

24  0 

9.6093 

28 

9.6486 

0.3514 

9.9607 

% 

0  66 

6 

10 

9.6121 

28 

9.6520 

0.3480 

9.9602 

SO 

I 

0.6 

20 

9.6149 

28 

9.6SS3 

34 
33 
34 

0.3447 

9.9596 

6 

40 

2 

3 

I  .  2 

1.8 

30 

9.6177 

28 

9.6587 

0.3413 

9.9S90 

5 

30 

40 

9.6205 

27 

9.6620 

0.3380 

9.9584 

5 

20 

4 

2.4 

so 

9.6232 

27 

9.6654 

33 

0.3346 

9.9579 

6 

10 

\ 

3.0 
3.6 

as  0 

9.6259 

9.6687 

0.3313 

9.9573 

0  6s 

9 

4.8 
5.4 

log  cos 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

/  0 

39 

38„ 

37 

36 

35 

34 

33 

32 

31 

30 

29 

38 

27 

I 

3.9 

3-8 

3-7 

3.6 

3.5 

3.4 

3.3 

3.2 

3. 

I  3.0 

2.9 

2.8 

2.7 

2 

7.8 

7.6 

7.4 

7.2 

7.0 

6.8 

6.e 

6.4 

6. 

2  6.0 

S.8 

5.6 

5.4 

3 

II. 7 

II. 4 

II. I 

10.8 

10. 5 

10.2 

9.S 

9.6 

9. 

3  9.0 

8.7 

8.4 

8.1 

4 

15.6 

IS. 2 

14.8 

14.4 

14.0 

13.6 

13.! 

12.8 

12. 

\  12.0 

11.6 

II. 2 

10.8 

S 

19. 5 

19.0 

18.5 

18.0 

17.5 

17.0 

16. s 

16.0 

IS. 

5  iS.o 

14.  S 

14.0  13. 5 
16.8  16.2 

6 

23.4 

22.8 

22.2 

21.6 

21.0 

20.4 

19. 8 

19.2 

18. 

3  18.0 

17.4 

7 

27.3 

26.6 

25.9 

25.2 

24.5 

23.8 

23.1 

22.4 

21. 

^  21.0 

20.3 

19.6  18.9 

8 

31.2 

30.4 

29.6 

28.8 

28.0 

27.2 

26. j{ 

25.6 

24. 

5  24.0 

23.2 

22.4  21.6 

9 

35. 1 

34-2 

33-3 

32.4 

31. 5 

30.6 

29.7 

28.8 

27. 

J  27. 026. 1 

25.2  24.3 

REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     483 
Logarithms  of  Thigonometbic  Functions 


log  Bin       d      log  tan 


dc     log  cot      log  cos      d 


iS    0 


9.62S9 
9.6286 


9.6313 
30  9.6340 
40    9 . 6366 


26    o 


SO 


so 
28    0 

10 


30    0 


9.6392 

9.641S 
9.6444 

9.6470 
9.649s 
9.6S21 

9.6546 
9.6370 
9.6595 

9.6620 
9.6644 
9.6668 

9 . 6692 
9.6716 
9.6740 

9.6763 
9.6787 
9.6810 

9.6833 
9.6856 
9.6878 

9.6901 
9.6923 
9.6946 

9.6968 
9.6990 


log  cos 


9.6687 
9.6720 

9.6752 
9.6785 
9.6817 

9.6850 
9.6882 
9.6914 

9.6946 
9.6977 

9 . 7009 

9.7040 
9.7072 

9.7103 

9.7134 
9.7165 
9.7196 

9.7226 
9.7257 
9.7287 

9.7317 
9.7348 
9.7378 

9.7408 

9-7438 
9.7467 

9.7497 
9.7S26 
9.7356 

9.758s 
9.7614 


log  cot 


dc 


0.3313 
0.3280 

0.3248 
0.321S 
0.3183 

0.3150 
0.3118 
0.3086 

0.3054 
0.3023 
0.2991 

o . 2960 
0.292& 
0.2897 

0.2866 
0.2835 
0.2804 

0.2774 
0.2743 
0.2713 

0.2683 
0.2652 
0.2622 

0.2392 
0.2562 
0.2533 

0.2503 
0.2474 
0.2444 

0.2415 
0.2386 


log  tan 


9.9573 
9.9567 

9.9361 
9.9SS3 
9-9549 

9-9543 
9-9S37 
9-9530 

9-9524 
9-9518 
9-9512 

9-9505 
9-9499 
9.9492 

9.9486 
9.9479 
0-9473 

9  -  9466 
9-9459 
9-9453 

9.9446 
9.9439 
9.9432 


,9425 
.9418 
.9411 


9-9404 
9-9397 
9-9390 

9-9383 
9-937S 


log  sin 


6S 


o  64 


0  63 

SO 


o  62 


o  61 


o  60 


6 

0.6 

1.2 

1.8 

2.4 
3.0 
3.6 


7 
0.7 
1.4 
2.1 

2.8 

3.5 

4-2 

4-9 
3-6 
6-3 


8 

0.8 
1.6 
2.4 

3.2 
4-0 
4-8 

3-6 
6-4 

7-2 


,13 

32 

31 

30 

29 

27 

26 

2S 

24 

23 

T 

3-3 

3-2 

3.1 

I 

3-0 

2.0 

2.7 

2.6 

2-5 

I 

2.4 

2.3 

2 

6.6 

6-4 

6.2 

2 

6.0 

S.8 

5-4 

S-2 

5-0 

2 

4-8 

4-6 

3 

9-9 

9-6 

9.3 

3 

9-0 

8.7 

8-1 

7.8 

7-5 

3 

7-2 

6-9 

4 

13-2 

12.8 

12.4 

4 

12.0 

II. 6 

10-8 

10.4 

10 -0 

4 

9-6 

9-2 

S 

I6-I! 

16.0 

IS.  5 

5 

15-0 

14-S 

13-5 

13.0 

12-3 

S 

12.0 

ir.s 

6 

19-8 

19.2 

18.6 

6 

18.0 

17.4 

16.2 

IS-6 

15.0 

6 

14-4 

13.8 

7 

23.1 

22.4 
25.6 

21.7 

'7 

21.0 

20.3 

18-9 

18.2 

17-5 

7 

16-8 

16. 1 

8 

26.4 

24.8 

8 

24.0 

23.2 

21.6 

20.8 

20.0 

8 

19-2 

1S.4 

9 

29-7 

28.8 

27.9 

9 

27.0 

26.1 

24-3 

23-4 

22.5 

9 

21-6 

20.7 

4-4 
6.6 


8.8 


15-4 
17.6 
19.8 


484       ELEMENTARY  MATHEMATICAL  ANALYSIS 

LOGABITHMS  OP  TSiaONOMBTKIC   FUNCTIONS 


log  sin 


log  tan 


dc 


log  cot 


log  COS 


30 

0 
10 

9.6990 
9.7012 

20 
30 
40 

9.7033 
9.70SS 
9.7076 

31 

SO 

0 
10 

9.7097 
9.7II8 
9.7139 

20 
30 
40 

9.7160 

9.7I8I 

9.7201 

32 

so 

0 

10 

9.7222 
9.7242 
9.7262 

20 
30 

40 

9.7282 
9.7302 
9.7322 

33 

so 
0 

10 

9.7342 
9.7361 
9.7380 

4 

20 
30 
40 

9.7400 
9.7419 
9.7438 

34 

so 

0 

10 

9.74S7 

9.7476 
9.7494 

20 
30 
40 

9.7SI3 
9.7S3I 
9.7SSO 

35 

so 

0 

9.7S68 
9.7586 

9.7614 
9.7644 

9.7673 
9.7701 
9.7730 

9.77S9 
9.7788 
9.7816 

9.7845 
9.7873 
9 . 7902 

9.7930 
9.7958 
9.7986 

9 . 8014 
9 .  8042 
9.S070 

9.8097 
9.812s 
9.8153 

9.8180 
9.8208 
9.823s 

9 .  8263 
9.8290 
9.8317 

9.8344 
9.8371 
9.8398 

9.8425 
9.8452 


0.2386 
0.2356 

0.2327 
0.2299 
0.2270 

0.2241 
0.2212 
0.2184 

O.21SS 
0.2127 
0.2098 

0.2070 
0.2042 
0.2014 

0.1986 
O.I9S8 
0.1930 

0.1903 

0.1875 
0.1847 

0.1820 
0.1792 
0.176s 

0.1737 
0.1710 
0.1683 

o.i6s6 
0.1629 
0.1602 

0.1575 
0.1548 


9.937s 
9.9368 

9.9361 
9.9353 
9.9346 

9.9338 
9.9331 
9.9323 

9.931s 
9.9308 
9.9300 

9.9292 
9.9284 
9.9276 

9.9268 
9.9260 
9.9252 

9.9244 
9.9236 
9.9228 

9.9219 
9.9211 
9.9203 

9.9194 
9.9x86 
9.9177 

9.9169 
9.9160 
9.9151 

9.9142 
9.9134 


o  60 


o  58 


0  56 


log  COS 


log  cot 


log  tan 


log  sin 


30 
3.0 
6-0 
9.0 

29 

2.9 
S.8 
8.7 

12.0 

11.6 

15.0 
18.0 

14-5 
17.4 

21.0 

20.3 

24.0 
27.0 

23.2 
26.1 

2S 

2.8 
5.6 

8.4 

11.2 
14.0 
16.8 

19.6 

4 
25.2 


27 

22 

I 

2.7 

2.2 

2 
3 

5.4 
8.1 

Vi 

4 

10.8 

8.8 

i 

13. S 

16.2 

11.0 
13.2 

7 

8 

18.9 
21.6 

IS. 4 
17.6 

9 

24.3 

19.8 

4.2 

6.3 
8.4 

0.5 


14-7 
16.8 
18.9 


20 

10 

I 

2.0 

1.0 

2 

4.0 

.1.8 

3 

6.0 

5.7 

4 

8.0 

7.6 

S 

10. 0 

9.5 

6 

12.0 

11.4 

7 

14.0 
16.0 

13.3 

8 

15.2 

9 

18.0 

17. 1 

7 
0.7 
1.4 


2.8 
3.5 

4.2 

4.9 
S.6 
6.3 


8 

0.8 
1.6 
2.4 

3.2 
4.0 
4.8 

5.6 
6.4 
7.2 

9 

0.9 


18 

1.8 
3.6 
5.4 

7.2 

9.0 

10.8 

12.6 
14.4 
16.2 


REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     485 

LOQAEITHMS  OF  TeIGONOMBTBIC   FUNCTIONS, 


log  sin 


log  tan 


dc 


log  cot 


log  cos 


3S     0 


30 
40 

so 
36    o 


30 

40 

so 

37  0 

10 

20 
30 
40 

SO 

38  o 


39     o 


9-7585 
9 . 7604 

9.7622 
9.7640 
9.7657 

9-7675 
9.7692 
9.7710 
\ 
9.7727 
9.7744 
9.7761 

9.7778 
9-7795 
9-7811 

9.7S28 
9-7844 
9.7861 

9-7877 
9.7893 
9.7910 

9.7926 
9.7941 
9.7957 

9.7973 
9.7989 
9.8004 

9.8020 
9.8035 
9.8050 

9.8066 
9.8081 


log  COS 


9.8452 
9.8479 

9.8506 
9.8533 
9.8559 

9-8586 
9-8613 
9-8639 

9.8666 

9.8692 
9.8718 

9.8745 
9.8771 
9.8797 

9.8824 
9-8850 
9-8876 

9.8902 
9.8928 
9.8954 

g.8980 
9 .  9006 
9.9032 

9-9058 
9.9084 
9.9110 

9.9135 
9.9161 
9.9187 

9.9212 
9.9238 


log  cot 


dc 


0.1548 

O.IS2I 
0.1494 

o .  1467 

O.I44I 

O.I414 
0.1387 
O.I36I 

0.1334 
0.1308 
0.1282 

O.I2S5 
0.1229 
0.1203 

O.II76 
Q.I150 
o. 1124 

0.1098 
0.1072 
0.1046 

0.1020 
0 . 0994 
0.0968 

0.0942 
0.0916 
0.0890 

0.086s 
0.0839 
0.0813 

p. 0788 
0.0762 


9-9134 
9-9125 

9-9I16 
9-9107 
9.9098 

9.9089 
9.9080 
9.9070 

9.9061 

9-9052 

9 .  9042 

9-9033 
9-9023 
9-9014 

9.9004 
9.8995 
9-8985 

9-8975 
9-8965 
9-8955 

9-8945 
9-8935 
9-8925 

9-891S 
9-8905 
9-8895 

9.8884 
9-8874 
9  -  8864 

9-8853 
9.8843 


log  tan 


log  sin 


o  55 


0  S4 
50 

40 
30 
20 


o   S3 


50 


9 
0.9 
1.8 
2-7 

3-6 

4-5 
5-4 

6-3 
7-2 
8.1 


4.4 
S-5 
6-6 

7.7 
8.8 
9-9 


27 1 

2 

7 

s 

4 

8 

I 

10 

8 

1.1 

5 

16 

2 

t8 

9 

21 

6 

24 

3 

26 

2.6 
S.2 
7.8 

:o.4 
13  o 
IS. 6 

18.2 
20.8 
23-4 


Z-S 

18 

2-5 

1.8 

S-0 

3.6 

7-5 

5.4 

10. 0 

7-2 

r2.5 

9-0 

15.0 

10.8 

17.5 

12.6 

20.0 

14.4 

22.5 

16.2 

17 

1-7 
3-4 
5-1 

6.8 
8-5 


II. 9 
13-6 
IS-3 


16 

1.6 
3.2 

4-8 

6-4 
8-0 
9-6 


IS 
l-S 
3-0 
4-5 

6.0 
7-S 
9-0 


486       ELEMENTARY  MATHEMATICAL  ANALYSIS 

LOQARITHMS   OF   TRIGONOMETRIC   FUNCTIONS 


log  Bin 


log  tan 


dc 


log  cot 


log  COS 


42    o 


43    o 


44    o 


50 
45    0 


9 . 8081 
9 . 8096 

9.8111 
9.812s 
9.8140 

9.81SS 
9.8169 
9.8184 

9.8198 
9.8213 
9.8227 

9.8241 
9.82SS 
9.8269 

9.8283 
9.8297 
9.8311 

9.8324 
9.8338 
9.8351 

9.836s 
9-8378 
9.8391 

9.840s 
9.8418 
9.8431 

9.8444 
9.84S7 
9 . 8469 

9.8482 
9.849s 


9.9238 
9.9264 

9.9289 
9.931S 
9.9341 

9 . 9366 
9.9392 
9.94*7 

9.9443 
9.9468 
9.9494 

9.9SI9 
9. 9544 
9.9S70 

9.9S9S 
9. 9621 
9.9646 

9.9671 
9.9697 
9.9722 

9.9747 
9.9772 
9.9798 

9.9823 
9.9S48 
9.9874 

9.9899 
9.9924 
9.9949 

9. 9975 
O . 0000 


0.0762 
0.0736 

0.07II 
0.068s 
o.o6s9 

0.0634 
o . 0608 
0.0583 

0.0SS7 

O.OS32  , 

0.0506 

0.0481 
0.0456 
0 . 0430 

0 . 040s 
0.0379 
0.0354 

o . 0329 
0 .  0303 
0.0278 

0.0253 
0.0228 

0.0202 

0.0177 

0.0IS2 

0.0126 

O.OIOI 

0.0076 
0.0051 

0.002s 

0 . 0000 


log  COS 


log  cot 


dc 


log  tan 


9.8843 
9.8832 

9.8821 
9.88IC 
9.8800 

9.8789 
9.8778 
9.8767 

9.8756 
9.8745 
9.8733 

g.8722 
9.87II 

9 . 8699 

9.8688 
9.8676 

9 .  8665 

9.8653 

9.8641 
9.8629 

0.8618 
9.8606 
9.8594 

9.8582 
9.8569 
9.8557 

9.8545 
0.8532 
9.8520 

9.8507 
9.8495 


0  49 
so 


o  48 

SO 

40 
30 


0  47 
50 

40 
30 
20 


0  46 

SO 

40 
30 

20 


log  sin 


26 

2.6 

S.2 
7.8 

10.4 
13.0 

IS. 6 

18.2 
20.8 
23.4 


25 

2.5 

so 

.7.5 

10. o 

12. S 

IS.O 

17.  s 

20.0 

22.  S 


15 

1. 5 

3.0 

4.5 

6.0 
7.5 
9.0 

10. s 
12.0 
13. 5 


14 

1.4 
2.8 
4.2 

5.6 
7.0 
8.4 

9.8 


1.0 
2.0 
3.0 

4.0 
5.0 

6.0 

7.0 
8.0 
9.0 


13 
1.3 
2.6 
3.9 

5.2 

6.5 
7.8 

9.1 
10.4 

II. 7 


4.4 

s.s 

6.6 

7.7 
8.8 
9.9 


12 

1.2 

1:1 
4.8 
6.0 

7.2 

■8.4 
9.6 
10.8 


REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA    487 


Natural  Trigonometric  Functions 


Deg. 

Radians 

n  Bin 

n  CSC 

n  tan 

n  cot 

n  sec 

n  cos 

o 

0 . 0000 

.000 

.000 

1. 000 

1. 00 

1.5708 

90 

I 

2 

3 

0.0I7S 
0.0349 
0.0524 

.017 
.035 

.052 

57.3 
28.7 
19. 1 

.017 
.035 
.052 

57.3 
28.6 
19. 1 

1. 000 

1. 001 
1. 001 

i.OO 

•  999 
.999 

I.SS33 
1.5359 
I.S184 

89 
88 
87 

4 
1 

0 . 0698 
0.0873 
0 . 1047 

.070 
.087 
.105 

14-3 
II. 5 
9.57 

.070 
.087 
.105 

14.3 
II-4 
9.51 

1.002 
1.004 
1.006 

.998 
.996 
.995 

1.5010 

86 
84 

7 
8 
9 

0.1222 
0.1396 
0.IS7I 

.122 
.139 
.156 

8.21 
7.19 
6.39 

.123 
.141 
.158 

8.14 
7.12 
6.31 

1.008 
1. 010 
1. 012 

.993 
.990 
.988 

1.4486 
I. 4312 
1.4137 

83 
82 
81 

10 

0.1745 

.174 

5. 76 

.176 

S.67 

1. 015 

.98s 

1.3963 

80 

11 

12 

13 

0.1920 
0 . 2094 
0.2269 

.191 
.208 

.225 

5. 24 
4.81 
4.45 

.194 
.213 
.231 

S.14 
4.70 
4.33 

1. 019 
1.022 
1.026 

.982 

•  978 

•  974 

1.3788 
1.3614 
1.3439 

79 

78 
77 

14 
IS 
i6 

0.2443 
0.2618 
0.2793 

.242 
.259 
.276 

3.63 

■.HI 
.287 

4.01 
3.73 
3.49 

1. 031 
1.035 
1.040 

■.III 
.961 

1.3265 
1.3090 
I.291S 

76 
75 
74 

11 
19 

0.2967 
0.3142 
0.3316 

.292 
.309 
.326 

3.42 
3.24 
3.07 

.306 

.325 
.344 

3.27 
3.08 
2.90 

1.046 
1. 051 
1.058 

•  956 
.946 

I. 2741 
1.2566 
1.2392 

73 
72 
71 

20 

U.3491 

.342 

2.92 

.364 

2.75 

1.064 

.940 

1.2217 

70 

21 
22 
23 

0.366s 
0.3840 
0.4014 

.358 

.375 
.391 

2.79 
2.67 
2.56 

.384 
.404 
.424 

2.61 
2.48 
2.36 

1. 071 
1.079 
1.086 

.934 
.927 
.921 

I . 2043 
I. 1868 
I . 1694 

69 
68 
67 

t 

24 

0.4189 
0.4363 
0.4538 

.407 
Ml 

2.46 
2.37 
2.28 

1^1 

.4S8 

2.25 
2.14 
2.05 

1.095 
1. 103 
1. 113 

.914 
.906 
.899 

1.1519 
I. 1345 
1.1170 

66 
64 

11 

29 

0.4712 
0.4887 
0.S061 

1P 
.485 

9.20 
2.13 
2.06 

.510 
.532 

.554 

1.96 

1.88 
1.80 

1. 122 
1 .133 
1. 143 

.891 
.883 
.875 

I . 0996 
1.0S21 
1 .  0647 

62 
61 

30 

0.5236 

.500 

2.00 

•  577 

1.73 

1. 155 

.866 

1.0472 

60 

31 
32 

33 

0.S4" 
0.558s 
0.5760 

.SIS 
.530 
.S4S 

1.94 
1.89 
1.84 

.601 
.625 
.649 

1.66 
1.60 
1. 54 

1.167 
1. 179 
1. 192 

.857 
.848 
.839 

1.0297 
1.0123 
0.9948 

59 
58 
57 

34 
35 
36 

0.5934 
0.6109 
0.6283 

.559 
.574 
.588 

1.79 
1. 74 
1.70 

.675 
.700 
.727 

1.48 
1-43 
1.38 

1.206 
1.221 
1.236 

.829 
.819 
.809 

0.9774 
0.9599 
0.942s 

56 
55 
54 

37 
38 
39 

0.6458 
0.6632 
0.6807 

.602 
.616 
.629 

1.66 
1.62 
1.59 

.754 
.781 
.810 

1-33 
1.28 
1.23 

1.252 
1.269 
1.287 

.799 
.788 
.777 

0.9250 
0.9076 
0.8901 

53 
52 

SI 

40 

0.6981 

.643 

x.s6 

.839 

X.19 

1.305 

.766 

0.8727 

SO 

41 
42 
43 

0.7156 
0.7330 
0.7505 

.656 
.669 
.682 

1.52 
1.49 
1.47 

.869 
.900 
•  933 

1. 15 

HI 

1.07 

1.32s 
1.346 
1.367 

.7SS 
■  743 
.731 

0.8S52 
0.8378 
0.8203 

49 
48 
47 

44 
45 

0.7679 
0.7854 

.69s 

.707 

1.44 
1. 41 

.966 
1. 00 

1.04 

1. 00 

1.390 

1.414 

.719 
.707 

0.8029 
0.7854 

46 

45 

n  cos 

n  sec 

n  cot 

n  tan 

n  CSC 

n  sin 

Radians 

Deg. 

488       ELEMENTARY  MATHEMATICAL  ANALYSIS 


Antilogakithms 


0    ]     I  Ia|3<4l5|6|7l8  19 

I  2  3  14  S  617  8  9l 

•SO 

•SI 
•S2 
■  S3 

3162 

3170 

3177 

3184 

3192 

3199 

3206 

3214 

3221 

3228 

112 

3  4  4 

5  6  7 

3236 
33II 
3388 

3243 
3319 
3396 

3251 
3327 
3404 

3258 
3334 
3412 

3266 
3342 
3420 

3273 
3350 
3428 

3281 
3357 
3436 

3289 
3365 
3443 

3296 
3373 
3451 

3304 
3381 
3459 

122 

12   2 
12   2 

3  4  5 
3  4  5 
3  4  5 

5  6  7 

5  6  7 

6  6  7 

.S4 

J 

is7 
.S8 
■  S9 

.6^ 

.6i 
.62 
.63 

.64 

it 

3467 
3548 
3631 

347S 
3SS6 
3639 

3483 
3565 
3648 

3491 
3573 
3656 

3499 
3581 
3664 

3S08 
3589 
3673 

3516 

3597 
3681 

3524 
3606 
3690 

3532 
3f4 
3698 

3540 
3622 
3707 

12   2 
12   2 

r  2  3 

3  4  5 
3,  4  5 
3  4  5 

6  6  7 

678 

37IS 
3802 
3890 

3724 
3811 
3899 

3733 
3819 
3908 

3828 
3917 

3750 
3837 
3926 

3758 
3846 
3936 

3767 
38SS 
3945 

3776 
3864 
3954 

3784 
3873 
3963 

3793 
3882 
3972 

I  2  3 
I  2   3 
123 

3  4  5 

4  4  5 
4  5   5 

i  '  I 

t  '  1 
678 

3981 

3990 

3999 

4009 

4018 

4027 

4036 

4046 

40SS 

4064 

I  2   3 

4  5  6 

678 

4074 
4169 
4266 

4083 
4178 
4276 

4093 
4188 
4285 

4102 
4198 
4295 

4HI 
4207 
4305 

4121 
42J7 
431s 

4130 
4227 
4325 

4140 
4236 
4335 

4IS0 
4246 
4345 

4159 
4256 
4355 

1  2  3 
I  2   3 
I  2   3 

456 
456 
4  5  6 

7   8  9 
7  8  9 
7   8  9 

436s 
4467 

4S7I 

437S 
4477 
4581 

438s 
4487 
4592 

4395 
4498 
4603 

4406 
4508 
4613 

4416 
4519 
4624 

4426 
4529 
4634 

4436 
4539 
4645 

4446 
4550 
4656 

4457 
4667 

I  2   3 
I  2   3 
I  2   3 

456 
456 
456 

7   8  9 
7   8  9 
7  9  10 

.69 

4677 
4786 
4898 

4688 
4797 
4909 

4699 
4808 
4920 

4710 
4819 
4932 

4721 
4831 
4943 

4732 
4842 
4955 

4742 
4853 
4966 

4'|3 
4864 
4977 

4764 
4875 
4989 

4775 
4887 
5000 

I  2   3 
I  2   3 
I  2  3 

4  5   7 
467 

5  6  7 

8  9  10 
8  9  lO 
8  9  10 

.70 

5012 

S023 

S035 

S047 

S058 

5070 

S082 

SO93 

Sios 

5117 

I  2   4 

S  6  7 

8  9  II 

.71 
.72 
.73 

.74 

S129 
S248 
S370 

S140 
5260 
S383 

5152 
5272 
5395 

5164 
S284 
S408 

5176 
5297 
5420 

S188 
S309 
5433 

5200 

5321 
S44S 

5212 
5333 

5458 

5224 
5346 
5470 

5236 
5358 
S483 

I  2   4 
I  2  -4 
I  3   4 

5  6  7 
5  6  7 
568 

8  10  II 

9  10  II 
9  10  II 

549S 
5623 
S7S4 

SS08 
S636 
S768 

5S2I 
5649 
5781 

5534 
5662 
5794 

5546 
5675 
5808 

SSS9 
S689 
5821 

5572 
5702 
5834 

5585 
S7I5 
5848 

5598 
5728 
S86i 

S6io 
5741 
587s 

1  3  4 
I  3  4 
134 

568 
5   7   8 
5   7   8 

9  10  12 
9  10  12 
9  II  12 

■77 
.78 
•79 

5888 
6026 
6166 

S902 

6039 
6180 

5916 
6053 
6194 

5929 
6067 
6209 

5943 
6081 
6223 

5957 
6095 
6237 

5970 
6109 
6252 

5984 
6124 
6266 

5998 
6138 
6281 

6012 
6152 
6295 

I  3  4 
I  3  4 
I  3  4 

5   7   8 
678 
679 

10  II  12 
10  II  13 
10  II  13 

.So 

6310 

6324 

6339 

6353 

6368 

6383 

6397 

6412 

6427 

6442 

I  3  4 

679 

10  12  13 

.81 
.82 
.83 

6457 
6607 
6761 

6471 
6622 
6776 

6486 
6637 
6792 

6501 
6653 
6808 

6si6 
6668 
6823 

6531 
6683 
6839 

6855 

6s6l 
6714 
6871 

6577 
6730 
6887 

6592 
674s 
6902 

235 
235 
235 

689 
689 
689 

II  12  14 
II  12  14 
II  13  14 

.85 

.86 

.i? 

.88 
.89 

.90 

6918 
7079 
7244 

6934 
7096 

7261 

6950 
7112 
7278 

6966 
7129 
7295 

6»82 
7145 
7311 

6998 
7161 
7328 

70IS 
7178 
7345 

7031 
7194 
7362 

7047 
7211 
7379 

7063 
7228 
7396 

235 
235 
235 

6  8  10 

7  8  10 
7  8  10 

11  13  15 

12  13  15 
12  13  15 

7413 
7S86 
7762 

7430 
7603 
7780 

7447 
7621 
7798 

7816 

7482 
7656 
7834 

7499 
7674 
7852 

7516 
7691 
7870 

7534 
7709 
7889 

7SSI 
7727 
7907 

7568 
7745 
7925 

235 
245 
245 

7  9  10 
7  9  II 
7  9  II 

12  14  16 

12  14  16 

13  14  16 

7943 

7962 

7980 

7998 

8017 

803S 

8054 

8072 

8091 

8110 

246 

7  9  II 

13  15  17 

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8318 
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8356 
8551 

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8375 
8570 

8204 
839s 
8590 

8222 

8241 
8433 
8630 

8260 
8453 
8650 

8279 
8472 
8670 

8299 
8492 
8690 

246 
246 
246 

8  9  II 
8  10  12 
8  10  12 

13  IS  17 

14  IS  17 
14  16  18 

■94 
.96 

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8710 
8913 
9120 

9333 
9SS0 
9772 

8730 
8933 

9141 

8750 
8954 
9162 

8770 
8974 
91S3 

8790 
8995 
9204 

8810 
9016 
9226 

8831 
9036 
9247 

8851 
9057 
9268 

8872 
9078 
9290 

8892 
909? 
931 1 

2  4  6 
246 
246 

8  10  12 
8  10  12 
8  II  13 

14  16  18 

15  17  19 
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9572 

9376 
9594 
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9397 
9616 
9840 

9419 
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9661 
0886 

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9683 
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9484 
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9506 
9727 
9954 

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247 
247 
2  5   7 

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9  II  13 

9  "  14 

15  17  20 

16  18  20 
16  18  20 

REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA     489 


Antilogamthms 


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■21 
■22 
•23 

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■25 
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■29 

1000 

1002 

1005 

1007 

1009 

1012 

1014 

1016 

1019 

IO21 

0    0       I 

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222 

1023 
1047 
1072 

1026 
lOSO 

1074 

102S 
1052 
1076 

1030 
1054 
1079 

1033 
1057 
1081 

1035 
1059 
1084 

1038 
1062 
1086 

1040 
1064 
1089 

1042 
1067 
IO91 

1045 
1069 
1094 

00     I 

0   0       I 

00  I 

I    I    I 

III 

222 
2  2  2 
2       2       2 

1096 
1122 
1 148 

1099 

II2S 

IISI 

1102 
I127 
1153 

IlO/i 

II30 
II56 

I107 
1132 
I159 

1 109 
II3S 
1161 

II12 
1138 
1 164 

II14 
I140 
1167 

III7 
1 143 
1 169 

II19 
I146 
I172 

0    1       I 

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oil 

112 
112 
112 

2  2  2 
2  2  2 
2       2       2 

1175 
1202 
1230 

II78 
1205 
1233 

1180 
1208 
1236 

II83 

I2II 
1239 

1 186 
1213 
1242 

I189 
1216 
1245 

I191 
1219 
1247 

1194 
1222 
1250 

II97 
1225 
1253 

I199 
1227 
1256 

0    I       I 

oil 
oil 

112 
112 
112 

2  2  2 
223 
2       2      3 

I2S9 

1262 

126s 

1268 

1271 

1274 

1276 

1279 

12S2 

128s 

0    1       I 

112 

223 

1288 
1318 
1349 

I29I 
I32I 
1352 

1294 
1324 
1355 

1297 

1327 
1358 

1300 
1330 
1361 

1303 

1334 
1365 

1306 
1337 
1368 

1309 
1340 
1371 

13I2 
1343 
1374 

1315 
1346 
1377 

oil 
oil 
oil 

12       2 
12       2 
12       2 

223 
223 
233 

1380 
1413 
1445 

1384 
I4I6 
1449 

1387 
1419 
1452 

1390 
1422 

I4SS 

1393 
1426 
1459 

1396 
1429 
1462 

1400 
1432 
1466 

1403 
1435 
1469 

1406 
1439 
1472 

1409 
1442 
1476 

0    1       I 
0    I       I 
0    I       I 

12       2 
12       2 
12       2 

233 
233 
233 

1479 
IS14 
IS49 

1483 
ISI7 

ISS2 

i486 
1521 
1556 

1489 
1524 
1560 

1493 
1528 
1563 

1496 
1531 
1567 

1500 
1535 
1570 

1503 
1538 
1574 

1507 
1542 
1578 

151O 
IS4S 
1581 

oil 

0    1       I 
0    1       I 

12       2 
12       2 
12       2 

2  '3  3 
233 

3  3      3 

ISSS 

IS89 

1592 

1596 

1600 

1603 

1607 

1611 

1614 

1618 

oil 

12       2 

3      3     3 

1622 
1660 
1698 

1626 
1663 

1702 

1629 
1667 
1706 

1633 
167 1 
1710 

1637 
1675 
1714 

1641 
1679 
1718 

1644 
1683 
1722 

1648 
1687 
1726 

1652 
1690 
1730 

1656 
1694 
1734 

0    I       I 
0    I       I 
0    1       I 

2       2       2 
2       2       2 
2       2       2 

3  3  3 
3  3^3 
3     3     4 

1738 
1778 
1820 

1742 
1782 
1824 

1746 
1786 
1828 

1750 
1791 
1832 

1754 
1795 
1837 

1758 
1799 
1841 

1762 
1803 
1845 

1766 
1807 
1849 

1770 
1811 
1854 

1774 
1816 
1858 

oil 
oil 

0    1       I 

2       2       2 
2       2       2 
223 

3  3  4 
3  3  4 
3     3     4 

1862 
190S 
1950 

1866 
I9I0 

1954 

1871 
1914 
1959 

1875 
1919 
1963 

1879 
1923 
1968 

1884 
1928 
1972 

1888 
1932 
1977 

1892 
1936 
1982 

1897 
1941 
1986 

1901 
I94S 
1991 

0    I        I 
0    I        I 

223 
223 
2       2       3 

3  3  4 
3  4  4 
3     4     4 

.30 

■31 
•32 
■33 

•34 
■35 
■36 

1995 

2000 

2004 

2009 

2014 

2018 

2023 

2028 

2032 

2037 

0    1        I 

223 

3     4     4 

2042 
2089 
2138 

2046 
2094 
2143 

2051 
2099 
2148 

2056 

2104 
2153 

2061 
2109 
2158 

2065 
2113 
2163 

2070 
2118 
2168 

2075 
2123 
2173 

2080 
2128 
2178 

2084 
2133 
2183 

01     I 

0    1        I 
0    I        I 

223 
223 
2    2    3 

3  4  4 
3  4  4 
3     4     4 

21S8 
2239 
2291 

2193 
2244 
2296 

2198 
2249 
2301 

2203 
2254 
2307 

2208 
2259 
2312 

2213 
2265 
2317 

2218 
2270 
2323 

2223 
2275 
2328 

2228 
2280 
2333 

2234 
2286 
2339 

112 
112 
112 

233 
233 
233 

4  4  5 
4  4  5 
4     4     5 

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■39 

.40 

2344 
2399 
2455 

2350 
2404 
2460 

2355 
2410 
2466 

2360 
2415 
2472 

2366 
2421 
2477 

2371 
2427 
2483 

2377 
2432 
2489 

2382 
2438 
249s 

2388 
2443 
2500 

2393 
2449 
2506 

112 
112 
l'   I       2 

233 

233 
233 

4  4  5 
4  4  5 
4     5      5 

2SI2 

2518 

2523 

2529 

2535 

2541 

2547 

2553 

2559 

2564 

112 

234 

4     5      5 

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.42 
.43 

•44 

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■.ti 
■49 

2570 
2630 
2692 

lilt 
2698 

2582 
2642 
2704 

2588 
2649 
2710 

2594 
2655 
2716 

2600 
2661 
2723 

2606 
2667 
2729 

25l2 
2673 
2735 

2618 
2679 
2742 

2624 
268s 
2748 

112 
112 
112 

234 

2  3     4 

3  3     4 

4  5  5 
456 
456 

2754 
2818 
2884 

2761 
2825 
2891 

2767 
2831 
2897 

2773 
2838 
2904 

2780 
2844 
2911 

2786 
28S1 
2917 

2793 
2858 
2924 

2799 
2864 
2931 

2805 
2871 
2938 

2812 
2877 
2944 

112 
I     I        2 
I     I       2 

3     3      4 
3     3     4 
3     3     4 

4  5     6 

5  5  S 
5      5     6 

29SI 
3020 
3090 

2958 
3027 
3097 

296s 
3034 
3105 

2972 
3041 
3112 

2979 
3048 
3119 

2985 
3055 
3126 

2992 
3062 
3133 

2999 
3069 
3141 

3006:3013 
30763083 
314813155 

112 
112 
112 

3     3     4 
3     4     4 
3     4     4 

5  5  6 
566 
566 

INDEX 


(The  numbers  refer  to  the  pages) 


Abscissa,  33 

Absolute  value  of  complex  num- 
ber, 369 
Addition  formulas  for  sine  and 
cosine,  307-309 
for  tangent,  309 
Additive   properties   of   graphs, 

142,  295-297 
Aggregation,  symbols  of,  453 
Algebraic  scale,  3,  357 
Alternating  current  curves,  384 
et  seq. 
represented    by    complex 
numbers,  384 
Amplitude  of  complex  number, 
369 
of  S.  H.  M.,  340 
of  sinusoid,  117 
of  uniform  circular  motion, 

102 
of  wave,  345 
Angle,  99 

depression,  130 
direction,  103 
eccentric,  156 
elevation,  130 
epoch,  340,  345 
initial  side,  99 
phase,  340,  345 
that  one  line  makes  with  an- 
other, 313 
vectorial,  103 
Angles,  congruent,  100 
Angular  magnitude,  99 
units  of  measure,  100 
velocity,  102 


Anti-logarithm,  254 
Approximation  formulas,  209 
Approximations,  successive,  196 
Argument  of  function,  12 

of  complex  number,  369 
Arithmetical  mean,  213 

progression,  213-216 

triangle,  204 
Asymptotes    of    hyperbola,    60, 

165,  167 
Auxiliary  circles,  155 
Axes  of  ellipse,  154 

of  hyperbola,  168 

Binomial   coefficients,    graphical 
representation  of,    211, 
212 
theorem,  204  et  seq. 
Briggs,  Henry,  236 

system  of  logarithms,  245 

Cartesian  coordinates,  33 

Cassinian  ovals,  394 

Catenary,  297 

Change  of  base,  264,  265 

of  unit,  66,  70,  foot  note,  77 
et  seq.,  285 

Characteristic,  250,  251 

Circle    and    circular    functions. 
Chap.  IV,  97  et  seq. 

Circle,  dipolar,  395 
equation  of,  97,  98 
sine  and  cosine,  126-128 
tangent  to,  422,  428 
through  three  points,  433 


491 


492 


INDEX 


(The  numbers  refer  to  the  pages) 


Circles,  auxiliary,  155 
Circular  functions,    103   et  seq. 
graphical    computation    of, 

106,  115 
fundamental  relations,   110, 

304-318 
law  of,  132 
motion,  102 
Cologarithm,  254 
Combinations,    199,   202,   Chap. 

VII 
Common  logarithms,  246 
Complementary  angles,  114,  118 
Completing  square,  463 
Complex   numbers,   Chap.    XII, 
357  et  seq. 
defined,  363 
laws  of,  365 
polar  form,  369 
typical  form,  363 
Composite  angles,  functions  of, 

310-312 
Composition  of  two  S.  H.  M.'s, 

343 

Compound  harmonic  motion,  334 

interest,  220 

law,  277 

Computers  rules,  328 

Conditional  equations,  138,  320- 

327 
Conies,  414,  417 
con-focal,  441 

sections.  Chap.  XIV,  399  et 
seq. 
Conjugate  axis,  168 

complex  numbers,  367 
hyperbolas,  170 
Connecting  rod  motion,  355 
Constants  and  variables,  15 
Continuous  function,  11 

compounding  of  interest,  278 


Coordinate  paper,  27,   124,  271, 

289 
Coordinates,  Chap.  II,  23  et  seq. 

Cartesian,  34 

orthogonal,  124 

polar,  123,  434 

rectangular,   33  et  seq. 

relation  of  polar  and  rectan- 
gular, 136,  434 
Cosecant,  103 
Cosine,  103 

curve,  117,  126 

law,  321      ■ 
Cotangent,  103 
Crest  of  sinusoid,  116 
Cubical  parabola,  52 
Cubic  equation,  192  et  seq. 
"Cut  and  Try,"  149 
Cycloid,  395 


Damped  vibrations,  299 
Damping  factor,  299 
Decreasing  function,  63 

geometrical  series,  221 
DeMoivres  theorem,  375 
Descartes,  Ren6,  34 
Diameter  of  any  curve,  440 

of  ellipse,  440 

of  parabola,  419 
Direction  angle,  103 
Directrix  of  ellipse,  402,  415 

of  hyperbola,  408,  415 

of  parabola,  408,  413 
Discontinuous  function,   13,   27, 

59 
Distance     of    point    from    line, 

426 
Distributive   law  of   multiplica- 
tion, 205,  365 
general,  456 


INDEX 


493 


CThe  numbers  refer  to  the  pages) 


Double  angle,  functions  of,  315 
scale,  4-9,  21,  22,  266-276 
of  algebraic  functions,  21 
of   logarithmic   functions, 
266-276 

"e,"  241,  245,  260,  277 
Eccentric  angle,  156 
Eccentricity  of  earth's  orbit,  403 

of  ellipse,  401 

of  hyperbola,  408 

of  parabola,  413,  415 
Ellipse,   52   et  seq.,   399  et  seq., 
Chaps.  V  and  XIV. 

axes  of,  154 

construction,  155,  158,  159 

directrices,  402,  415 

eccentricity,  402 

focal  radii,  399,  430 

foci,  399 

latus  recturn,  404 

parametric  equation,  155 

polar  equation,  410 

symmetrical  equation  of,  154. 

tangent  to,  429 

vertices,  154 
Ellipsograph,  158 
Elliptic  motion,  344,  388 
Empirical  curves,  46,  283,  291 

formulas,  75 
Envelope,  422 

Epicycloid  and  epitrochoid,  397 
Epoch  angle,  340,  345,  348 
Equations,  conditional,  138 

explicit,  154 

quadratic,  462 
systems,  186-192 

simple,  461 

single     and     simultaneous, 
Chap.  VI,  174 

with  given  roots,  178 


Even  function,  119 
Expansion,  binomial,  205,  208 
Exponential   curves,    236-240, 
260-264 
equation,  240 

function,  Chap.  IX,  234  et  seq. 
compared    with    power, 

286-289 
defined,  240,  243,  244 
sums  of,  295-299 
Exponents,  definition  of,  466 
irrational,  243,  244 
laws  of,  466 

Factor  theorem,  177 
Factorial  number,  200 
Factoring,  454-460 

fundamental  theorem  in,  459 
Family  of  curves,  78 

of  lines,  421 
Focal  radii  and  foci,  393 
of  eUipse,  399,  430 
of  hyperbola,  406 

radius  of  parabola,  414 
Fractions,  460 
Frequency  of  S.  H.  M.,  340 

of  sinusoidal  wave,  347 

uniform  circular  motion,  102 
Function,  periodic,  26,  113 

power,  48  et  seq.,  286 

S.  H.  M.,  339 

trigononietric,  103 
Functions,  10,  11 

circular,    Chap.    IV,    97    et 
seq.,  103 

continuous,  11 

discontinuous,  13,  27,  59 

even  and  odd,  119 

explicit  and  implicit,  155, 174 

exponential,   240,-243,   244, 
286  et  seq. 


494 


INDEX 


(The  numbers  refer  to  the  pages) 


Functions,    increasing    and    de- 
creasing, 63,  147 

General  equation  of  second  de- 
gree, 440 
Geometrical  mean,  217 

progression,  217  et  seq. 
Graphical  computation,  16  et  seq. 
of  circular  functions,  106 
of  integral  powers,  19 
of  logarithms,  237 
of  product,  16 
of  quotient,  17 
of  sq.  roots,  18,  21 
of  squares,  18,  21 
solution  of  cubic,  192 

simultaneous      equations, 
183  et  seq. 
Graph   of  binomial  coefficients, 
211,  212 
of  complex  number,  364 
of  cycloid,  396 
of  ellipse,  158,  159 
of  equation,  36 
of  functions  of  multiple  an- 
gles, 318,  319 
of  geometrical  series,  236 
of  hyperbola,  167,  168 
of  hyperbolic  functions,  297 
of  logarithmic  and  exponen- 
tial curves,  236-240,  260 
of  parabolic  arc,  420 
of  power  function,  48-60,  64 
of  sinusoid,  115 
of  tangent  and  secant  curves, 
143-147 
Graphs,  suggestions  on  construc- 
tion of,  27 
nonnstatistical,  35 

Half-angle,  functions  of,  315  . 
Halley's  law,  282 


Haridonic  analysis,  354 

functions,  346 

fundamental,  352 

motion,  Chap.  XI,  339  et  seq. 
compound,  352 
Hyperbola,  Chap.  V  and  XIV. 

asymptotes,  165,  167 

axes,  168 

center,  168 

conjugate,  170 

construction  of,  167 

eccentricity,  408 

foci  and  focal  radii,  406 

latus  rectum,  408 

parametric    equations,    165, 
167 

polar  equation,  410 

rectangular,  58,  164 

symmetrical  equation,  166 

vertices,  168 
Hyperbolic  curves,  52,  58 

sine  and  cosine,  296 

system  of  logarithms,  245 
Hypocycloid  and  Hypo-trochoid, 
397 

i  =  V^^,  362 

Identities,  110,  111,  138,  304r-317 
Illustrations  from  science,  69-76 
Image  of  curve,  57 
Increasing  function,  63,  147 

progression,  214 
Increment,  logarithmic,  279 
Infinite  discontinuity,  59 

geometrical  progression,  221 
Infinity,  69 
Intercepts,  39,  40 
Interest,  compound,  220,  277 

curve,  237 
Interpolation,  252 
Intersection  of  loci,  92,  182 


INDEX 


495 


(The  numbers  refer  to  the  pages) 


Inverse  of  curve,  136 

of  straight  line  and  circle,  136 
trigonometric  functions,  137, 
360 

Irrational  numbers,  379 

Lamellar  motion,  88 

Langley's  law,  74 

Latitude  and  longitude  of  a  point, 
33 

Latus  rectum  of  ellipse,  404 
of  hyperbola,  408 
of  parabola,  414 

Law  of  circular  functions,  132 
of  complex  numbers,  365 
of  compound  interest,  277 
of  exponential  function,  288 
of  power  function,  80-82 
of  sines,   cosines,   and  tan- 
gents, 320-327 

Lead  or  lag,  349,  384 

Legitimate  transformations,   178 

Lemniscate,  393 

Limit,  221 

Limiting  lines  of  ellipse,  161 

Loci,  Chap.  XIII,  387  et  seq. 
defined  by  focal  radii,  393 
Theorems  on,  61,  62,  65,  85, 
88,  135 

Locus  of  points,  35,  36 
of  equation,  36 

Logarithmic     and     exponential 
functions,     Chap.     IX, 
234  et  seq. 
coordinate  paper,  289-295 
curves,  236-240,  260-264 
double  scale,  266 
functions,  240,  244 
increment    and    decrement, 

279-282,  299 
tables,  252,  253 


Logarithm  of  a  number,  236,  244 
Logarithms,  common,  244 
graph,  237-243 
properties  of,  247-250 
systems  of,  245 

Mantissa,  250 

Mean,  arithmetical,  213 

geometrical,  217 

harmonical,  224 
Modulus  of  complex  number,  369 

of  decay,  281,  299 

of  logarithmic  system,  264 
Motion,  circular,  102 

compound  harmonic,  352 

connecting  rod,  355 

elliptic,  344,  388 

shearing,  87 

S.  H.  M.,  339  et  seq. 

Naperian  base,  245,  260,  277,  341 

system  of  logs.,  245 
Napier,  John,  234 
Natural  system  of  logarithms,  245 
Negative  angle,  100 

functions  of,  118,  119 
Newton's  law,  282 
Node,  116 
Normal,  136 

equation  of  line,  136,  423 

to  ellipse,  430 

to  parabola,  420 

Oblique  triangles,  320-334 
Odd  functions,  119 
Operators,  359 
Ordinate  of  point,  33 
Origin,  34 

at  vertex,  160,  415 
Orthogonal  systems,  124 
Orthographic     projection,    120- 
123,  152,  265 


496 


INDEX 


(The  numbers  refer  to  the  pages) 


Paper,   logarithmic,    289   et  seq. 
polar,  124  et  seq. 
rectangular,  33  et  seq. 
semi-log,  271,  283  et  seq. 
Parabola,  52,  413 
cubical,  52 
polar  equation,  414 
properties  of,  419 
semi-cubical,  52 
Parabolic  curves,  49  et  seq.,  56,  289 
Parameter,  155,  387 
Parametric  equations,  155 
of  cycloid,  396 
of  ellipse,  155 
of  hyperbola,  165,  166 
Pascal's  triangle,  204,  205 
Periodic     functions     {see    trig.- 

fcns.),  26,  116 
Period  of  S.  H.  M.,  341 

of  simple  pendulum,  342 
of   uniform   circular   mo- 
tion, 102 
of  wave,  347 
Permutations,  199-202 

and     combinations,     Chap. 
VII,  198  et  seq. 
Phase  angle,  341,  348,  349 
Plane  triangles,  320-334 
Polar  coordinates,  123,  434 

diagrams   of   periodic   func- 
tions, 126,  318 
equation  of  ellipse,  410 
of  hyperbola,  410 
of  parabola,  414 
of  straight  hne,  135 
form  of  complex  number,  369 
relation  to  rectangular,  136, 
434 
Polynomial,  175 

Positive  and  negative  angle,  100, 
119 


Positive    and    negative    coordi- 
nates, 33 
side  of  line,  427 
Power  function.  Chap.  Ill,  48  et 
seq. 
compared  with   exponen- 
tial, 286-289 
law  of,  80-82 
practical  graph,  76 
variation  of,  62 
Probability  curve,  212 
Products,  special,  451,  452 
Progressions,  Chap.  VIII,  213  et 
seq. 
arithmetical,  213-216 
decreasing,  214 
geometrical,  217-224 
harmonical,  224,  225 
Projection,    orthographic,     120- 

123,  152,  265 
Proportionality  factor,  68 

Quadrants,  34 
Quadratic  equations,  462 

systems  of  equations,  186 
Questionable    transformations, 
178 

Radian  unit  of  measure,  101,  102 

Radicals,  reduction  of,  471 

Radius  vector,  123 

Ratio  definition  of  conies,  414, 415 

Rationalization,  472 

Rational  formulas,  75 
numbers,  354 

Rectangular  coords,  (see  Coordi- 
nates), Chap.  II,  33 
et  seq. 

Reflection  of  curve,  57 

Remainder  theorem,  175 

Reversors,  361 

Right  angle  system,  100 


INDEX 


497 


(The  numbers  refer  to  the  pages) 


Root  of  any  complex  number,  376 
of  equation,  91 
of  function,  91,  177 
of  utoity,  377 

Rotation  of  locus,  82,  133 

polar  coordinates,  133-135 
rectangular,  434^436 
of  rigid  body,  82 

Scalar  numbers,  358 
Scale,  1,  3 

algebraic,  3,  357 

functions,  21 
arithmetical,  3,  357 
double,  4  et  seq. 

logarithmic,  266-276 
uniform,  1 

Tables,  damped  vibrations,  301, 
302 
logarithms,  252,  476 
natural  trig,  functions,  107, 

128,  129,  487 
powers,  51 
of  "e,"  263 
Tangent,  103 
graph,  143 
law,  323 

to  circle,  422,  428 
to  curve,  260 
to  ellipse,  429 
to  parabola,  418 
Theorems,  binomial,  205  et  seq. 
factor,  176 

functions  of  composite  an- 
gles, 310 
on  loci,  61,  65,  85,  135 
remainder,  175 
Transformations,  legitimate  and 

questionable,  178 
Translation,  82,  83 

of  any  locus,  83,  85,  425 
32 


Translation  of  rigid  body,  82 
Transverse  axis,  168 
Triangle  of  reference,    103,    108 
Triangles,  solution  of,  129,  320- 
338 
oblique,  320-338 
right,  129-131 
Trigonometric  curves,   115,   117, 
143-147,  319 
functions,  103  et  seq. 
Trochoid,  397 
Trochoidal  waves,  349 
Trough  of  sinusoid,  116 

Uniform  circular  motion,  102 
Unit,  change    of,   66,   70,    77  et 
seq.,  285 
of  angular  measure,  101 

Variables  and  constants,  15 

and  functions  of  variables. 
Chap.  I 
Variation,  67 

of  power  function,  62 
Vector,  123 

radius,  123 
Vectorial  angle,  103,  123 
Velocity,  angular,  102,  339 

of  wave,  348 
Versors,  362 
Vertices  of  ellipse,  154 

of  hyperbola,  168 
Vibrations,  damped,  299 

Waves,  Chap.  XI,  339  et  seq. 
compound,  352 
length  of,  347 
progressive,  344  et  seq. 
sinusoidal,  344  et  seq. 
stationary,  350 
trochoidal,  349 

Zero  of  function,  91 


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